Design of Smart Power Grid Renewable Energy Systems [3rd ed.] 1119573327, 9781119573326

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DESIGN OF SMART POWER GRID RENEWABLE ENERGY SYSTEMS

DESIGN OF SMART POWER GRID RENEWABLE ENERGY SYSTEMS Third Edition

ALI KEYHANI

This third edition first published 2019 © 2019 John Wiley & Sons, Inc. All rights reserved. Edition History John Wiley & Sons, Inc. (1e, 2011 and 2e, 2017) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Ali Keyhani to be identified as the author of this work has been asserted in accordance with law. Registered Office(s) John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Keyhani, Ali, 1942– author. Title: Design of smart power grid renewable energy systems / Ali Keyhani. Description: Third edition. | Hoboken, NJ : Wiley, 2020. | Includes bibliographical references and index. | Identifiers: LCCN 2019016024 (print) | LCCN 2019020733 (ebook) | ISBN 9781119573210 (Adobe PDF) | ISBN 9781119573340 (ePub) | ISBN 9781119573326 (hardback) Subjects: LCSH: Smart power grids–Textbooks. | Smart power grids–Design and construction–Textbooks. | Electric power systems–Automatic control–Textbooks. | Distributed generation of electric power–Computer simulation–Textbooks. | Renewable energy sources–Textbooks. | Electric circuits–Textbooks. | Electricity–Textbooks. Classification: LCC TK1007 (ebook) | LCC TK1007 .K49 2020 (print) | DDC 621.319/1–dc23 LC record available at https://lccn.loc.gov/2019016024 Cover design by Wiley Cover image: © Jason Winter/Shutterstock Set in 10/12pt TimesTen by SPi Global, Pondicherry, India Printed in the United States of America 10

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I dedicate this book to my parents, Dr. Mohammed Hossein Keyhani and Mrs. Batool Haddad

CONTENTS PREFACE

xiii

ACKNOWLEDGMENTS

xvi

ABOUT THE COMPANION WEBSITE

xvii

1

ENERGY AND CIVILIZATION

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17

1

Introduction: Motivation / 1 Fossil Fuel / 2 Energy Use and Industrialization / 2 Nuclear Energy / 4 Global Warming / 5 The Age of the Electric Power Grid / 9 Green and Renewable Energy Sources / 10 Hydrogen / 11 Solar and Photovoltaic / 11 1.9.1 Wind Power / 12 1.9.2 Geothermal / 13 Biomass / 13 Ethanol / 13 Energy Units and Conversions / 13 Estimating the Cost of Energy / 17 New Oil Boom–Hydraulic Fracturing (Fracking) / 20 Estimation of Future CO2 / 21 The Paris Agreement | UNFCCC / 22 Energy Utilization and Economic Growth / 23 vii

viii

CONTENTS

1.18 Conclusion / 23 Problems / 24 Further Reading / 26 2

POWER GRIDS

28

2.1 2.2

Introduction / 28 Electric Power Grids / 29 2.2.1 Background / 29 2.2.2 The Construction of a Power Grid System / 29 2.3 Basic Concepts of Power Grids / 33 2.3.1 Common Terms / 33 2.3.2 Calculating Power Consumption / 33 2.4 Load Models / 49 2.5 Transformers in Electric Power Grids / 53 2.5.1 A Short History of Transformers / 54 2.5.2 Transmission Voltage / 54 2.5.3 Transformers / 55 2.6 Modeling a Microgrid System / 59 2.6.1 The Per Unit System / 60 2.7 Modeling Three-Phase Transformers / 69 2.8 Tap-Changing Transformers / 72 2.9 Modeling Transmission Lines / 74 Problems / 87 References / 92 3 3.1 3.2 3.3

3.4 3.5

3.6 3.7 3.8

3.9

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS 93 Introduction / 93 Single-Phase DC/AC Inverters with Two Switches / 94 Single-Phase DC/AC Inverters with a Four-Switch Bipolar Switching Method / 106 3.3.1 Pulse Width Modulation with Unipolar Voltage Switching for a Single-Phase Full-Bridge Inverter / 110 Three-Phase DC/AC Inverters / 113 Pulse Width Modulation Methods / 114 3.5.1 The Triangular Method / 114 3.5.2 The Identity Method / 119 Analysis of DC/AC Three-Phase Inverters / 120 Microgrid of Renewable Energy Systems / 130 DC/DC Converters in Green Energy Systems / 133 3.8.1 The Step-Up Converter / 134 3.8.2 The Step-Down Converter / 144 3.8.3 The Buck–Boost Converter / 151 Rectifiers / 156

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CONTENTS

3.10 3.11

Pulse Width Modulation Rectifiers / 160 A Three-Phase Voltage Source Rectifier Utilizing Sinusoidal PWM Switching / 163 3.12 The Sizing of an Inverter for Microgrid Operation / 167 3.13 The Sizing of a Rectifier for Microgrid Operation / 169 3.14 The Sizing of DC/DC Converters for Microgrid Operation / 170 Problems / 171 References / 176 4

SMART POWER GRID SYSTEMS

177

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

Introduction / 177 Power Grid Operation / 178 Vertically and Market-Structured Power Grid / 184 The Operations Control of a Power Grid / 187 Load Frequency Control / 187 Automatic Generation Control / 193 Operating Reserve Calculation / 198 Basic Concepts of a Smart Power Grid / 199 The Load Factor / 206 The Load Factor and Real-Time Pricing / 209 A Cyber-Controlled Smart Grid / 212 Smart Grid Development / 214 Smart Microgrid Renewable and Green Energy Systems / 216 4.14 A Power Grid Steam Generator / 223 4.15 Power Grid Modeling / 234 Problems / 240 References / 245 5

SOLAR ENERGY SYSTEMS

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12

Introduction / 247 The Solar Energy Conversion Process: Thermal Power Plants / 251 Photovoltaic Power Conversion / 253 Photovoltaic Materials / 253 Photovoltaic Characteristics / 255 Photovoltaic Efficiency / 258 The Design of Photovoltaic Systems / 262 The Modeling of a Photovoltaic Module / 277 The Measurement of Photovoltaic Performance / 278 The Maximum Power Point of a Photovoltaic Array / 278 A Battery Storage System / 292 A Storage System Based on a Single-Cell Battery / 294

247

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CONTENTS

5.13

The Energy Yield of a Photovoltaic Module and the Angle of Incidence / 317 5.14 The State of Photovoltaic Generation Technology / 318 Problems / 318 References / 326 6

MICROGRID WIND ENERGY SYSTEMS

328

6.1 6.2 6.3 6.4

Introduction / 328 Wind Power / 329 Wind Turbine Generators / 331 The Modeling of Induction Machines / 334 6.4.1 Calculation of Slip / 343 6.4.2 The Equivalent Circuit of an Induction Machine / 343 6.5 Power Flow Analysis of an Induction Machine / 346 6.6 The Operation of an Induction Generator / 351 6.7 Dynamic Performance / 366 6.8 The Doubly Fed Induction Generator / 372 6.9 Brushless Doubly Fed Induction Generator Systems / 375 6.10 Variable-Speed Permanent Magnet Generators / 376 6.11 A Variable-Speed Synchronous Generator / 377 6.12 A Variable-Speed Generator with a Converter Isolated from the Grid / 378 Problems / 380 References / 384 7

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

7.1 7.2 7.3 7.4 7.5 7.6 7.7

7.8 7.9 7.10 7.11 7.12 7.13 7.14

386

Introduction / 386 Voltage Calculation in Power Grid Analysis / 387 The Power Flow Problem / 391 Load Flow Study as a Power System Engineering Tool / 392 Bus Types / 392 General Formulation of the Power Flow Problem / 397 Algorithm for Calculation of Bus Admittance Model / 400 7.7.1 The History of Algebra, Algorithm, and Number Systems / 400 7.7.2 Bus Admittance Algorithm / 402 The Bus Impedance Algorithm / 403 Formulation of the Load Flow Problem / 404 The Gauss–Seidel YBUS Algorithm / 407 The Gauss–Seidel ZBUS Algorithm / 412 Comparison of the YBUS and ZBUS Power Flow Solution Methods / 419 The Synchronous and Asynchronous Operation of Microgrids / 420 An Advanced Power Flow Solution Method: The Newton–Raphson Algorithm / 422 7.14.1 The Newton–Raphson Algorithm / 425

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CONTENTS

7.15 General Formulation of the Newton–Raphson Algorithm / 430 7.16 The Decoupled Newton–Raphson Algorithm / 434 7.17 The Fast Decoupled Load Flow Algorithm / 435 7.18 Analysis of a Power Flow Problem / 436 Problems / 448 References / 461 8

POWER GRID AND MICROGRID FAULT STUDIES

462

8.1 8.2 8.3 8.4 8.5 8.6

Introduction / 462 Power Grid Fault Current Calculation / 464 Symmetrical Components / 468 Sequence Networks for Power Generators / 473 The Modeling of Wind and PV Generating Stations / 476 Sequence Networks for Balanced Three-Phase Transmission Lines / 477 8.7 Ground Current Flow in Balanced Three-Phase Transformers / 479 8.8 Zero Sequence Network / 481 8.8.1 Transformers / 481 8.8.2 Load Connections / 482 8.8.3 Power Grid / 484 8.9 Fault Studies / 487 8.9.1 Balanced Three-Phase Fault Analysis / 490 8.9.2 Unbalanced Faults / 508 8.9.3 Single-Line-to-Ground Faults / 508 8.9.4 Double-Line-to-Ground Faults / 511 8.9.5 Line-to-Line Faults / 513 Problems / 527 References / 533 9

SMART DEVICES AND ENERGY EFFICIENCY MONITORING SYSTEMS

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13

Introduction / 534 Kilowatt-Hour Measurements / 535 Current and Voltage Measurements / 536 Power Measurements at 60 or 50 HZ / 537 Analog-to-Digital Conversions / 538 Root Mean Square (RMS) Measurement Devices / 538 Energy Monitoring Systems / 539 Smart Meters / 539 Power Monitoring and Scheduling / 540 Communication Systems / 541 Network Security and Software / 543 Smartphone Applications / 546 Summary / 546

534

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ENERGY SAVING AND COST ESTIMATION

Problems / 547 Further Reading / 548 10 LOAD ESTIMATION AND CLASSIFICATION

549

10.1 10.2 10.3

Introduction / 549 Load Estimation of a Residential Load / 549 Service Feeder and Metering / 557 10.3.1 Assumed Wattages / 557 Problems / 560 References / 562 11 ENERGY SAVING AND COST ESTIMATION OF INCANDESCENT AND LIGHT-EMITTING DIODES

563

11.1 11.2

Building Lighting with Incandescent Bulbs / 563 Comparative Performance of LED, Incandescent, and LFC Lighting / 564 11.3 Building Load Estimation / 566 11.4 Led Energy Saving / 569 11.5 Return on Investment on LED Lighting / 571 11.6 Annual Carbon Emissions / 572 Problems / 572 References / 572 APPENDIX A COMPLEX NUMBERS

573

APPENDIX B TRANSMISSION LINE AND DISTRIBUTION TYPICAL DATA

576

APPENDIX C ENERGY YIELD OF PHOTOVOLTAIC PANELS AND ANGLE OF INCIDENCE

581

APPENDIX D WIND POWER

594

INDEX

599

PREFACE Sustainable energy production and efficient utilization of available energy resources, thereby reducing or eliminating our carbon footprint, are some of our greatest challenges in the twenty-first century. This is a particularly perplexing problem for those of us in the discipline of electrical engineering. This book addresses the problem of sustainable energy production as part of the design of microgrid and smart power grid renewable energy systems. The book also presents the design of microgrid photovoltaic (PV) power plants for residential and commercial buildings. Today the Internet offers vast resources for engineering students. As instructors, it is our job to provide a learning path for utilizing these resources. We should challenge our students with problems that attract their imagination. This book addresses this task by providing a systems approach to the global application of the presented concepts in sustainable green energy production, as well as analytical tools to aid in the practical design of renewable microgrids. In each chapter, the book presents an engineering problem and then formulates a mathematical model of the problem followed by a simulation testbed in MATLAB®, highlighting solution steps. Numerous solved examples are presented, and design problems to challenge the student are given at the end of each chapter. The book provides an Instructor’s Solution Manual and PowerPoint lecture notes with animation that can be adapted and changed, as instructors deem necessary for their presentation styles. Many examples utilizing MATLAB are presented to teach students in coding M-Files for solving engineering problems. Homework problems are presented at the end of each chapter. The book website address is www.wiley.com/go/smartpowergrid3e xiii

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PREFACE

The concepts presented in this book integrate three areas of electrical engineering: power systems, power electronics, and electric energy conversion systems. The book also addresses the fundamental design of wind and PV energy microgrids as part of smart bulk power grid systems. A prerequisite for the book is a basic understanding of electric circuits. The book builds its foundation by introducing phasor systems, three-phase systems, transformers, loads, DC/DC converters, DC/AC inverters, and AC/DC rectifiers, which are all integrated into the design of microgrids for renewable energy as part of bulk interconnected power grids. In the first chapter, in addition to a historical perspective of energy use, an analysis of fossil fuel use is provided through a series of calculations of carbon footprints. In Chapter 2, we review the basic principles underlying power systems, single-phase loads, three-phase loads, single- and three-phase transformers, distribution systems, transmission lines, and power system modeling. The generalized per unit system of power system analysis is also introduced. In Chapter 3, the topics include AC/DC rectifiers, DC/AC inverters, DC/DC converters, and pulse width modulation (PWM) methods. The focus is on the utilization of inverters as a three-terminal element of power systems for the integration of wind and PV energy sources; MATLAB simulations of PWM inverters are also provided. In Chapter 4, the fundamental concepts in the design and operation of smart grid power grids are described. This chapter introduces the power grid elements and their functions from a systems approach and provides an overview of the complexity of smart power grid operations. Topics covered in the chapter include the basic system concepts of sensing, measurement, integrated communications, and smart meters; real-time pricing; cyber control of intelligent grids; high green energy penetration into the bulk interconnected power grids; intermittent generation sources; and the electricity market. We are also introduced to the basic modeling and operation of synchronous generator operations, the limit of power flow on transmission lines, power flow problems, load factor calculations and their impact on the operation of smart grids, real-time pricing, and microgrids. These concepts set the stage for the integration of renewable energy in electric power systems. Chapter 5 presents PV energy sources. We learn how to compute the energy yield of PV modules and the angle of inclination for modules concerning their position to the sun for maximum energy yields. The chapter also presents the modeling of PV modules, the microgrid design of PV power plants, and the maximum power point tracking of PV systems. Chapter 6 introduces wind power generation by describing the modeling of induction generators as part of a microgrid of renewable energy systems. In this chapter, we also study the utilization of doubly fed induction generators, variable-speed permanent magnet generators, brushless generators, and variable-speed wind power conversion from a system’s perspective. In Chapter 7, the modeling bus admittance and bus impedance for power grids are presented, as well as a power flow analysis of microgrids as part of interconnected bulk power systems,

PREFACE

xv

followed by a simulation testbed in MATLAB for Gauss–Seidel, Newton– Raphson, and fast decoupled power flow studies. In Chapter 8, we study the resolving of three-phase quantities into their zero, positive, and negative sequence. Zero sequence equivalent circuits for different transformer connections are shown. Further, symmetric three-phase fault calculations and different types of asymmetric fault calculations are discussed. In Chapters 9 and 10, load estimation, smart devices, and energy monitoring are presented. In Chapter 11, the design PV microgrid of residential and commercial buildings by a design example is presented. This book provides the fundamental concepts of power grid integration on microgrids of green energy sources that are on the technology roadmap for virtually all nations. The design of microgrids is the driver for the modernization of infrastructure using green energy sources, power electronics, control, sensor technology, computer technology, and communication systems. ALI KEYHANI September 21, 2018 New York City

ACKNOWLEDGMENTS Over the years, many graduate and undergraduate students have contributed to the material presented in this book, in particular Chris Zuccarelli, Abir Chatterjee, and Ehsan Dadashnialehi of the Ohio State University, Paloma Sodhi of IIT India, Vefa Karakasli of Istanbul Technical University, and Adel El Shahat Lotfy Ahmed of the Department of Electrical Engineering of Georgia Southern University. I would like to acknowledge Hossein Torkaman, Associate Professor of Electrical Engineering, and Muhammad Reza Arabshahi at Shahid Beheshti University, and Mr. Edwin Lim, Engineering Computer Services at the Ohio State University. I would also like to thank Mr. Brett Kurzman, Mr. Elisha Benjamin, and Amudhapriya Sivamurthy, Production Editor, K&L Content Management at Wiley.

xvi

ABOUT THE COMPANION SITE This book is accompanied by a companion website:

www.wiley.com/go/smartpowergrid3e The website includes materials for students and instructors: Instructors • • • •

PowerPoint presentations for Chapters 1–11 PowerPoint presentations on selected control topics Projects Solution manuals for Chapters 1–11

Students • PowerPoint presentations for Chapters 1–11 • PowerPoint presentations on selected control topics xvii

CHAPTER 1

ENERGY AND CIVILIZATION 1.1

INTRODUCTION: MOTIVATION

Energy technology plays a central role in societal, economic, and social development. Fossil fuel-based technologies have advanced our quality of life, but at the same time, these advancements have come at a very high price. Fossil fuel sources of energy are the primary cause of environmental pollution and degradation; they have irreversibly destroyed aspects of our environment. Global warming is a result of our fossil fuel consumption. For example, the fish in our lakes and rivers are contaminated with mercury, a byproduct of rapid industrialization. The processing and use of fossil fuels have escalated public health costs: Our health care dollars have been and are being spent on treating environmental pollution-related health problems, such as black lung disease in coal miners. Our relentless search for and need to control these valuable resources have promoted political strife. We are now dependent on an energy source that is unsustainable as our energy needs grow and we deplete our limited resources. As petroleum supplies dwindle, it will become increasingly urgent to find energy alternatives that are sustainable as well as safe for the environment and humanity.

Design of Smart Power Grid Renewable Energy Systems, Third Edition. Ali Keyhani. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/smartpowergrid3e

1

2

ENERGY AND CIVILIZATION

1.2

FOSSIL FUEL

Fossil fuels—oil, natural gas, and coal—formed in Earth around 300 million years ago. Over millions of years, the decomposition of flora and fauna remains that lived in the world’s oceans produced the first oil. As the oceans receded, these remains were covered by layers of sand and earth and were subjected to severe climate changes: the Ice Age, volcanic eruption, and drought burying them even deeper in the Earth’s crust and closer to the Earth’s core. From the intense heat and pressure, the remains essentially were boiled into the oil. If you check the word, “petroleum” in a dictionary, you find it means “rock oil” or “oil from the earth.” The ancient Sumerians, Assyrians, Persians, and Babylonians found oil at the bank of the Karun and Euphrates rivers as it seeped above ground. Historically, humans have used oil for many purposes. The ancient Persians and Egyptians used liquid oil as a medicine for wounds. The Zoroastrians of Iran made their fire temples on top of percolating oil from the ground. Native Americans used oil to seal their canoes. Up to the fifteenth century, history of humanity’s energy use was limited. Regardless we can project the impact of energy on early civilizations from artifacts and monuments. The legacy of our oldest societies and their use of energy in the form of wood, wood charcoal, wind, and water power can be seen in the pyramids of Egypt, the Parthenon in Greece, the Persepolis in Iran, the Great Wall of China, and the Taj Mahal in India.

1.3

ENERGY USE AND INDUSTRIALIZATION

Figure 1.1 depicts the approximate time needed to develop various energy sources. Coal, oil, and natural gas fuels take millions of years to form. The oil that is consumed today was created more than a million years ago in the Earth’s crust. Our first energy source was wood. Then wood charcoal and coal replaced wood, and oil began to replace some of our coal usages to the point that oil and gas now supply most of our energy needs. Since the Industrial Revolution, we have used coal. Since 1800, for approximately 200 years, we have been using oil. However, our first energy source was wood and wood charcoal, which we used to cook food. Recorded history shows that humanity has been using wood energy for 10,000 years. In the near future we will exhaust oil and gas reserves. Oil and gas are not renewable: we must conserve energy and save our oil—and gas as well. Figure 1.2 depicts the world’s oil production (consumption) from 1965 to 2000 and estimated from 2005 to 2009. US oil production peaked around 1970. However, by using the fracking technology, oil production in the United States has rapidly increased (Section 1.14). Europe’s oil production is limited except for the North Sea oil reserve; it depends entirely on oil production from other parts of

Time

ENERGY USE AND INDUSTRIALIZATION

3

Millions of years

Month/years

Weeks

Hours/days Direct

Energy production Oil

Biomass

Hydroelectric power

Wind energy

Solar heat/ electricity

Figure 1.1 The approximate time required for the production of various energy sources.

9

×104

World oil production (Thousand barrels daily)

8

7

6

5

4

3 1965

1970

1975

1980

1985

1990

1995

2000

2005

2010

Year 1973: October War

1979: Iranian Revolution

Figure 1.2 The world’s oil production (consumption) from 1965 to 2000 and estimated from 2005 to 2009. Source: Based on Figure TS.2 from Solomon et al.

4

ENERGY AND CIVILIZATION

the world. In Asia, China, India, Japan, and Korea depend on imported oil. The rapid economic expansion of China, India, and Brazil are also rapidly depleting the world oil reserves. The Middle East has one of the largest oil reserves in the world. If the world reserves are used at the same rate as we do today, oil may run out in 40–100 years. Our natural gas reserves can be depleted in less than 60 years, and the coal reserves will be exhausted in 200 years. The impact of fossil fuel, coal, oil, and gas is manifested in climate change and the rise of temperature and sea level around the world. All can be said about the future is “Some things are so unexpected that no one is prepared for them.” Predicting the future is a fool’s game. However, we can empower every energy user to become an energy producer. We can develop a new energy economy based on renewable sources and create technology for conserving energy, reducing carbon footprints, and improving the quality of life on the planet. We can develop a distributed renewable energy system for every community and eliminate the long periods of blackout. We can make every energy user an energy producer by use of the solar and wind energy (https://en.wikipedia.org/wiki/Reservesto-production_ratio). Students are encouraged to study up-to-date information on the world energy consumption and production from the Wikipedia (https://en. wikipedia.org/wiki/Energy_development#Fossil_fuels).

1.4

NUCLEAR ENERGY

In 1789, Martin Heinrich Klaproth, a German chemist, discovered uranium while studying the mineral pitchblende. Eugène-Melchior Péligot, a French chemist, was the first person to isolate the metal, but it was Antoine-César Becquerel, a French physicist, who recognized its radioactive properties almost 100 years later. In 1934, Enrico Fermi used nuclear fuel to produce steam for the power industry. Then, he participated in building the first nuclear weapon used in World War II. The US Department of Energy estimates that worldwide uranium resources are generally considered to be sufficient for at least several decades. The amount of energy contained in a mass of hydrocarbon fuel such as gasoline is substantially lower in much less mass of nuclear fuel. This higher density of nuclear fission makes it an essential source of energy; however, the fusion process causes additional radioactive waste products. The radioactive products remain for a long time, giving rise to a nuclear waste problem. The counterbalance to a low carbon footprint of fission as an energy source is the concern about radioactive nuclear waste accumulation and the potential for nuclear destruction in a politically unstable world.

GLOBAL WARMING

1.5

5

GLOBAL WARMING

Greenhouse gases in the Earth’s atmosphere emit and absorb radiation. This radiation is within the thermal infrared (IR) range. Since the burning of fossil fuel and the start of the Industrial Revolution, greenhouse gases have accumulated in atmosphere. The greenhouse gases are primarily water vapor, carbon dioxide, carbon monoxide, ozone, and some other gases. Within the atmosphere of Earth, greenhouse gases are trapped. Figure 1.3 depicts the process of solar radiation incident energy and reflected energy from the Earth’s surface and the Earth’s atmosphere. The solar radiation incident energy as depicted by circle 1 emitted from the sun and its energy is approximated as 343 W/m2. Some of the solar radiation, represented by circles 2 and 4, is reflected from the Earth’s surface and the Earth’s atmosphere. The total reflected solar radiation is approximated as 103 W/m2. Approximately 240 W/m2 of solar radiation, depicted by circle 3, penetrates through the Earth’s atmosphere. About half of the solar radiation (circle 5), approximately 168 W/m2, is absorbed by the Earth’s surface. This radiation (circle 6) is converted into heat energy. This process generates IR radiation in the form of the emission of a long wave back to Earth. A portion of the IR radiation is absorbed. Then, it is re-emitted by the greenhouse molecules trapped in the Earth’s atmosphere. Circle 7 represents the IR radiation. Finally, some of the IR radiation (circle 8) passes through the atmosphere and into space. As the use of fossil fuel is accelerated, the carbon dioxide in the Earth’s atmosphere is also accelerated. The World Meteorological Organization (WMO) is the international body for the monitoring of climate change. The WMO has clearly stated

Sun 2 1

4 8

Atmosphere 3

Greenhouse gases

5

6

7

Earth’s surface

Figure 1.3 The effects of sun radiation on the surface of the Earth.

6

ENERGY AND CIVILIZATION

360

Proportion of CO2 (ppm)

350 340 330 320 310 300 290 280 270 1700

1800

1900

2000

Year

Figure 1.4 The production of CO2 since 1700. Source: Based on Figure TS.2 from Solomon et al.

the potential environmental and socioeconomic consequences for the world economy if the current trend continues. In this respect, global warming is an engineering problem, not a moral crusade. Until we take serious steps to reduce our carbon footprints, pollution and the perilous deterioration of the environment continue. The Intergovernmental Panel on Climate Change (IPCC) is the leading international body for the assessment of climate change. It was established by the United Nations Environment Programme (UNEP) and the WMO in 1988 to provide the world with a clear scientific view on the current state of knowledge in climate change and its potential environmental and socioeconomic impacts. In the same year, the UN General Assembly endorsed the action by WMO and UNEP in jointly establishing the IPCC. Figure 1.4 depicts the growth of carbon dioxide in our atmosphere in parts per million. Figure 1.5 depicts the condition of CO2 in the upper atmosphere. The Y-axis represents the magnitude of response. The X-axis is plotted to show the years into the future. The Y-axis, showing response efforts, does not have units. The CO2 emission into the atmosphere has peaked during the last 100 years. If concentrated efforts are made to reduce the CO2 emission and it is reduced over the next few hundred years to a lower level, the Earth’s temperature will continue to rise. Figure 1.6 depicts the stabilization of CO2 over the subsequent centuries. The reduction of CO2 reduces its impact on the Earth’s atmosphere; nevertheless, the existing CO2 in the atmosphere continues to raise the Earth’s temperature by a few tenths of a degree. Figure 1.7 depicts temperature stabilization after reduction of CO2 emission.

CO2 emission

Magnitude of response

CO2 emission Peak 0 –100 years

Today 100 years

1000 years

Figure 1.5 The effect of carbon dioxide concentration on temperature and sea level.

CO2 stabilization

Magnitude of response

Today 100 years

1000 years

Figure 1.6 CO2 stabilization has been achieved.

Temperature stabilization

Magnitude of response

Today 100 years

1000 years

Figure 1.7 Temperature stabilization after reduction of CO2 emission.

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ENERGY AND CIVILIZATION

Sea level rise due to thermal expansion

Magnitude of response

Today 100 years

1000 years

Figure 1.8 The sea level rise after the reduction of CO2.

Sea level rise due to ice melting

The rise in the temperature due to trapped CO2 in the Earth’s atmosphere impacts the thermal expansion of oceans. Figure 1.8 depicts the Earth’s surface temperature stabilization over a few centuries. As the ice sheets continue to melt due to rising temperatures over the next few centuries, the sea level also continues to rise. Figure 1.9 depicts the sea level rise after the reduction of CO2 in the atmosphere. As a direct consequence of trapped CO2 in the atmosphere, with its melting of the polar ice caps causing increased sea levels that bring coastal flooding, our pattern of life on Earth will be changed forever. Magnitude of response

Today 100 years

1000 years

Figure 1.9 The sea level rise after the reduction of CO2 in the atmosphere.

THE AGE OF THE ELECTRIC POWER GRID

1.6

9

THE AGE OF THE ELECTRIC POWER GRID

Hans Christian Ørsted, a Danish physicist and chemist, discovered electromagnetism in 1820. Michael Faraday, an English chemist and physicist, worked for many years to convert electrical force into magnetic force. In 1831, Faraday’s many years of effort were rewarded when he discovered electromagnetic induction; later, he invented the first dynamo and the first generator, a simple battery as a source of DC power simple battery. In 1801, an Italian physicist, Alessandro Giuseppe Antonio Anastasio Volta, invented the chemical battery. The second crucial technological development was the discovery of Faraday’s law of induction. Faraday is credited with the invention of the induction phenomenon in 1831. However, recognition for the induction phenomenon is also accorded to Francesco Zantedeschi, an Italian priest and physicist in 1829, and around the 1830s to Joseph Henry, an American scientist. Nikola Tesla was the main contributor to the technology on which electric power is based and its use of alternating current. He is also known for his pioneering work in the field of electromagnetism in the late nineteenth and early twentieth centuries. Tesla put world electrification in motion. By the 1920s, electric power production using fossil fuels to generate the electricity had started around the world. Since then, electric power has been used to power tools and vehicles; to provide heat for residential, commercial, and industrial systems; and to provide our energy needs in our everyday lives. Figure 1.10 shows the US production of electric power from 1920 to 1999. The International Energy Agency (IEA) forecasts an average annual growth rate of 2.5% for world electricity demand. At the rate around 2.5%, the world electricity demand will double by 2030. The IEA forecasts that the world carbon dioxide emissions due to power generation will increase by 75% by 2030. In 2009, the world population was approximately 6.8 billion. The United Nations forecasts population growth to 8.2 billion by 2030. Without interventions to contain population growth, another 1.5 billion people will need electric power equivalent to five times the current US rate of electrical power consumption. Figure 1.10 also projects that we can slow the growth of electric power production from fossil fuels by replacing them with renewable sources and integrating the green energy sources in electric power grids. As more countries such as China, India, Brazil, Indonesia, and others modernize their economy, the rate of CO2 production accelerates. Figure 1.11 shows the mean smooth recorded temperature by the UNEP. We can only hope that we can stop the trend of global warming as presented in Figure 1.11.

ENERGY AND CIVILIZATION

US electricity net generation (thousand kilowatt-hours)

10

5

×107 Plugging vehicles

4.5 4

If supplied by green energy

3.5 Smart grid: PV and wind

3 2.5 2 1.5 1 0.5 0 1900

1920

1940

1960

1980

2000

2020

2040

Year

Figure 1.10

0.8 0.6 Degrees C

0.4

The US production of electric power from 1920 to 1999.

1 Projected temperature rise without reduction of greenhouse gases

2 Projected temperature stabilization with reduction of greenhouse gases

0.2 0 –0.2 –0.4 –0.6 –0.8

1840 1860 1880 1900 1920 1940 1960 1980 2000 2020 2040 Year

Figure 1.11 The smooth average of published records of surface temperature from 1840 to 2000.

1.7

GREEN AND RENEWABLE ENERGY SOURCES

To meet carbon reduction targets, it is important we begin to use sources of energy that are renewable and sustainable. The need for environmentally friendly methods of transportation and stationary power is urgent. We need to replace traditional fossil-fuel-based vehicles with electric cars and the stationary power from traditional fuels, coal, gas, and oil with green sources for sustainable energy fuel for the future.

SOLAR AND PHOTOVOLTAIC

1.8

11

HYDROGEN

Besides renewable sources, such as wind and the sun, hydrogen (H) is an important source of clean, renewable energy. It is abundantly available in the universe. It is found in small quantities in the air. It is nontoxic, colorless, and odorless. Hydrogen can be used as an energy carrier, stored, and delivered to where it is needed. When hydrogen is used as a source of energy, it gives off only water and heat with no carbon emissions. Hydrogen has three times as much energy for the same quantity of oil. A hydrogen fuel cell is fundamentally different from a hydrogen combustion engine. In a hydrogen fuel cell, hydrogen atoms are divided into protons and electrons. The negatively charged electrons from hydrogen atoms create an electrical current with water as a by-product (H2O). Hydrogen fuel cells are used to generate electric energy at stationary electric power generating stations for residential, commercial, and industrial loads. The fuel cell can also be used to provide electrical energy for an automotive system, that is, a hydrogen combustion engine. Hydrogenbased energy has the potential to become a significant energy source in the future, but many related technical problems must be solved for the construction of a new infrastructure for future energy systems.

1.9

SOLAR AND PHOTOVOLTAIC

Solar and photovoltaic (PV) energy are also important renewable energy sources. The sun, the Earth’s primary source of energy, emits electromagnetic waves. It has invisible IR (heat) waves, as well as light waves. IR radiation has a wavelength between 0.7 and 300 micrometers (μm) or a frequency range between approximately 1 THz (terahertz; 10 to the power of 12) and 430 THz. Sunlight is defined by irradiance, meaning radiant energy of light. The solar irradiance at 1 peak sun-hour is equal to 1 kWh/m2. We represent one sun as the brightness to provide an irradiance of about 1 kilowatt-hour (kWh) per square meter (m2) at sea level and 0.8 suns about 800 Wh/m2. One sun’s energy has 523 watts of IR light, 445 watts of visible light, and 32 watts of ultraviolet (UV) light. Example 1.1 Compute the area in meter per square and square feet needed to generate 5000 kWh of power. Assume the sun irradiance is equivalent to 0.8 sun of energy. Solution Power capacity of PV at 0.8 sun = 0.8 kWh/m2 Capacity in kWh = (sun irradiance in kWh/m2) × (required area in m2) Required area in m2 = 5000 kWh/0.8 kWh/m2 = 6250 m2 1 m2 = 10.764 ft2 Required area in ft2 = (6250) ∙ (10.764) = 67,275 ft2

12

ENERGY AND CIVILIZATION

Plants, algae, and some species of bacteria capture light energy from the sun, and through the process of photosynthesis, they make food (sugar) from carbon dioxide and water. As the thermal IR radiation from the sun reaches the Earth, some of the heat is absorbed by the Earth’s surface, and some heat is reflected into space (see Figure 1.4). Highly reflective mirrors can be used to direct thermal radiation from the sun to provide a source of heat energy. The heat energy from the sun—solar thermal energy—can be used to heat water to a high temperature and pressurized conventionally to run a turbine generator. Solar PV sources are arrays of cells of silicon materials that convert solar radiation into direct current electricity. The cost of a crystalline silicon wafer is very high, but new light-absorbent materials have significantly reduced the cost. The most common elements are amorphous silicon (a-Si), mainly O for p-type Si and C, and the transition metals, mainly Fe. These materials are silicon put into different forms or polycrystalline materials, such as cadmium telluride (CdTe) and copper indium (gallium) (CIS and CIGS). The front of the PV module is designed to allow maximum light energy to be captured by the Si materials. Each cell generates approximately 0.5 V. Normally, 36 cells are connected in series to provide a PV module producing 12 V. Example 1.2 Compute the area in meter per square and square feet needed to generate 1000 kW of power. Assume the sun irradiant is equivalent to 0.4 sun of energy. Solution Power capacity of PV at 0.4 sun = 0.4 kWh/m2 Required area in m2 = 1000/0.4 = 2500 m2 1 m2 = 10.764 ft2 Required area in ft2 = (2500) ∙ (10.764) = 26,910 ft2 1.9.1

Wind Power

Wind is developed by uneven heating of water and land, which causes the flow of air. Therefore, wind is air in motion. As the sun rises, the air over the land heats up faster than the air over water. The heated air above the land swells and rises, and the denser, colder air flows rapidly to take the place of heated air and in the process generating wind. However, during the night, the process reverses. A wind turbine has blades and captures the wind’s kinetic energy. The blades are connected to a drive shaft that turns an electric generator to produce electricity. Students are encouraged to obtain up-to-date information on the wind energy production and cost from https://www.eia.gov and http://energy. gov/eere/wind/about-doe-wind-program and NREL (https://www.nrel.gov). NREL is an excellent source for the updated commercialization of renewable energy sources.

ENERGY UNITS AND CONVERSIONS

1.9.2

13

Geothermal

Renewable geothermal energy refers to the heat produced deep under the Earth’s surface. It is found in hot springs and geysers that come to the Earth’s surface or in reservoirs deep beneath the ground. The Earth’s core is made of iron surrounded by a layer of molten rocks or magma. Geothermal power plants are built on geothermal reservoirs, and the energy is primarily used to heat homes and commercial industry in the area. 1.10 BIOMASS Biomass is a type of fuel that comes from organic matter like agricultural and forestry residue, municipal solid waste, or industrial waste. The organic matter used may be trees, animal fat, vegetable oil, rotting waste, and sewage. Biofuels, such as biodiesel fuels, are currently mixed with gasoline for fueling cars or are used to produce heat or as fuel (wood and straw) in power stations to produce electric power. Rotting waste and sewage generate methane gas, which is also a biomass energy source. However, some controversial issues surrounded the use of biofuel. Producing biofuel can involve cutting down forests, transforming the organic matter into energy can be expensive with higher carbon footprints, and agricultural products may be redirected instead of being used for food. 1.11 ETHANOL Another source of energy is ethanol, which is produced from corn and sugar as well as other means. However, the analysis of the carbon cycle and the use of fossil fuels in the production of “agricultural” energy leaves many open questions: per year and unit area solar panels produce 100 times more electricity than corn ethanol. As we conclude this section, we need always to remember the Royal Society of London’s 1662 motto: “Nullius in verba” (Take Nobody’s Word). 1.12 ENERGY UNITS AND CONVERSIONS To estimate the carbon footprint of different classes of fossil fuels, we need to understand the energy conversion units. Because fossil fuels are supplied from different sources, we need to convert to equivalent energy measuring units to evaluate the use of all sources. The energy content of different fuels is measured by the heat that can be generated. One British thermal unit (Btu or BTU) requires 252 calories; it is equivalent to 1055 joules. The joule (J) is named after James Prescott Joule (born December 24, 1818), an English physicist and brewer who discovered the relationship between heat and mechanical work, which led to the fundamental theory of the conservation of energy.

14

ENERGY AND CIVILIZATION

TABLE 1.1 Carbon Footprint of Various Fossil Fuels for Production of 1 kWh of Electric Energy CO2 Footprint (lb/kWh)

Fuel Type Wood Coal-fired plant Gas-fired plant Oil-fired plant Combined-cycle gas

3.306 2.117 1.915 1.314 0.992

TABLE 1.2 Carbon Footprint of Green and Renewable Sources for Production of 1 kWh of Electric Energy Fuel Type

CO2 Footprint (lb/kWh)

Hydroelectric PV Wind

0.0088 0.2204 0.03306

The Btu or BTU is a traditional unit of heat. One BTU of heat raises one pound of water one degree Fahrenheit ( F). Heat is also now known to be equivalent to energy. In the International System of Units (SI), energy is measured in joule; one BTU is about 1055 joules. For measurement of a large amount of energy, the term “quad” is used. A quad is a unit of energy equal to 1015 BTU or 1.055 × 1018 joules. From your first course in physics, you may recall that one joule in the metric system is equal to the force of one newton (N) acting through one meter (m). Thus one joule is equal to one newton (N) times one meter (m) (1 J = 1 N × 1 m)—one watt equal to one joule per second. Therefore, one joule is the amount of work required to produce one watt of power for one second. In the BTU’s per hour unit, 3.41 Btu/h is equal to 1 W, and 1 BTU/h is equal to 0.2930 W (Tables 1.1 and 1.2). Example 1.3 Compute the amount of energy in Wh needed to bring 100 lb of water from 0 to 212 F. Solution Heat required = (100 lb) 212 F = 21,200 BTU Energy in Wh = (1055) ∙ (21,200)/3600 = 6212.77 Wh For direct current (DC) electricity P=V I where I represents the current through the load and V is the voltage across the load and unit of power, P, is in watts if the current is in amperes (A) and voltage in volts. Therefore, one kilowatt is a thousand watts. The energy

ENERGY UNITS AND CONVERSIONS

15

consumption is expressed in kilowatt-hour (kWh). One kWh is the energy consumed by a load for one hour. kWh can also be expressed in joules, and one kilowatt-hour (kWh) is equal to 3.6 million joules. Recall from your introduction to chemistry course that one calorie (cal) is equal to 4.184 J. Therefore, it follows that thousand BTU is equal to 0.239 kWh—one MWh to 3.41 million BTU. Because power system generators are running on natural gas, oil, or coal, we express the energy from these types of fuel in kilowatts per hour. For example, one thousand cubic feet of gas (Mcf ) can produce 301 kWh, and one hundred thousand BTU can produce 29.3 kWh of energy. Example 1.4 10 kWh.

Compute the amount of heat in BTU needed to generate

Solution One watt = one joule/second (j/s) 1000 W = 1000 j/s 1 kWh = 1 × 60 × 60 × 1000 = 3600 kj/s 10 kWh = 36,000 kj/s One BTU = 1055.058 j/s Heat in BTU needed for 10 kWh = 36,000,000/1055.058 = 34,121.3 BTU The energy content of coal is measured in BTU produced. For example, a ton of coal can generate 25 million BTU: equivalently, it can generate 7325 kWh. Furthermore, one barrel of oil (i.e. 42 gallons) can produce 1700 kWh. Other units of interest are a barrel of liquid natural gas having 1030 BTU and one cubic foot of natural gas having 1030 BTU. Example 1.5

Compute how many kWh is produced from 10 tons of coal.

Solution One ton of coal = 25,000,000 BTU 10 ton of coal = 250,000,000 BTU 1 kWh = 3413 BTU Energy used in kWh = (250,000,000)/3413 = 73,249.3 kWh Example 1.6 Compute the CO2 footprint of a residential home using 100 kWh of coal for one day. Solution One kWh of electric energy using a coal-fired plant produces 2.117 lb of CO2 (https://www.eia.gov/tools). Residential home carbon footprint for 100 kWh = (100) ∙ (2.117) = 211.7 lb of CO2

16

ENERGY AND CIVILIZATION

The carbon footprint can also be estimated by carbon (C) rather than CO2. The molecular weight of C is 12 and CO2 is 44. (Add the molecular weight of C, 12, to the molecular weight of O2, 16 times 2 = 32, to get 44, the molecular weight of CO2.) The emissions expressed in units of C are converted to emissions in CO2. The ratio of CO2/C is equal to 44/12 = 3.67. Thus, CO2 = 3.67 C. Conversely, C = 0.2724 CO2. Example 1.7 Compute the CO2 (carbon footprint) of a residential home using 100 kWh using coal per day. Solution One kWh of electric energy using a coal-fired plant has 2.117 lb of CO2. Residential home carbon footprint for 100 kWh = (100) ∙ (2.117) = 211.7 lb. The carbon footprints of coal are the highest among fossil fuels. Therefore, coal-fired plants produce the highest output rate of CO2 per kWh. The use of fossil fuels also adds other gases to the atmosphere per unit of heat energy as shown in Table 1.3. We can also estimate the carbon footprints for various electrical appliances corresponding to the method used to produce electrical energy. For example, one hour’s use of a color television produces 0.64 pounds (lb) of CO2 if coal is used to generate the electric power. For coal, this coefficient is approximated to be 2.3 lb CO2/kWh of electricity. Example 1.8 A light bulb is rated 60 W. If the light bulb is on for 24 hours, how much electric energy is consumed? Solution The energy consumed is given as Energy consumed = (60 W) × (24 h)/(1000) = 1.44 kWh

TABLE 1.3 Fossil Fuel Emission Levels in Pounds per Billion Btu of Energy Input Pollutant

Natural Gas

Carbon dioxide Carbon monoxide Nitrogen oxides Sulfur dioxide Particulates Mercury

117,000 40 92 1 7 0.000

Oil

Coal

164,000 33 448 1,122 84 0.007

208,000 208 457 2,591 2,744 0.016

Source: Based on data from the US Energy Information Administration (EIA) (April 1999). Natural Gas 1998: Issues and Trends. http://webapp1.dlib.indiana.edu/virtual_disk_library/index. cgi/4265704/FID1578/pdf/gas/056098.pdf.

ESTIMATING THE COST OF ENERGY

Example 1.9

17

Estimate the CO2 footprint of a 60 W bulb on for 24 hours.

Solution Carbon footprint = (1.44 kWh) × (2.3 lb CO2/kWh) = 3.3 lb CO2 Large coal-fired power plants are highly economical if their carbon footprints and damage to the environment are overlooked. In general, a unit cost of electricity is an inverse function of the unit size. For example, for a 100 kW unit, the unit cost is $0.15/kWh for a natural gas turbine and $0.30/kWh for PV energy. Therefore, if environmental degradation is ignored, the electric energy produced from fossil fuel is cheaper based on the present price of fossil fuel. For a large coal-fired power plant, the unit of electrical energy is in the range of $0.04–$0.08/kWh. (Up-to-date information can be obtained from websites of NREL [https://www.nrel.gov] and EIA [http://www.eia.doe.gov].) Green energy technology needs supporting governmental policies to promote electricity generation from green energy sources. Economic development in line with green energy policies is required for reducing the ecologic footprint of a developing world. Ancient air bubbles trapped in ice enable us to step back in time and see what Earth’s atmosphere, and climate, were like in the distant past. They tell us that levels of carbon dioxide (CO2) in the atmosphere are higher than they have been at any time in the past 400,000 years. During ice ages, CO2 levels were around 200 parts per million (ppm), and during the warmer interglacial periods, they hovered around 280 ppm (see fluctuations in the graph). In 2013, CO2 levels surpassed 400 ppm for the first time in recorded history. This recent rise in CO2 shows a remarkably constant relationship with fossil-fuel burning based on the simple premise that about 60 percent of fossil-fuel emissions stay in the air. (https://climate.nasa.gov/climate_resources/24/graphic-the-relentlessrise-of-carbon-dioxide)

After thousands of years of burning wood and wood charcoal, the CO2 concentration was at 288 parts per million by volume (ppmv) in 1850 just at the dawn of the Industrial Revolution. By the year 2000, CO2 had risen to 369.5 ppmv, an increase of 37.6% over 250 years. The exponential growth of CO2 is closely related to the production of electric energy (see Figures 1.3 and 1.5). 1.13 ESTIMATING THE COST OF ENERGY As we discussed, the cost of electric energy is measured by the power used over time. The power demand of any electrical appliance is inscribed on the appliance and included in its documentation or its nameplate. However, the power consumption of an appliance is also a function of the applied voltage

18

ENERGY AND CIVILIZATION

and operating frequency. Therefore, the manufacturers provide on the nameplate of an appliance, the voltage rating, the power rating, and the frequency. For a light bulb, which is purely resistive, the voltage rating and power rating are marked on the light bulb. A light bulb rated at 50 W and 120 V means that if we apply 120 volts to the light bulb, 50 watts of energy is consumed. Again, energy consumption is expressed as P=V I

(1.1)

where the unit of power consumption, that is, P, is in watts. The unit of V is in volts and unit I is in amperes. The rate of energy consumption can be written as P=

dW dt

(1.2)

We can then write the energy consumed by loads (i.e. electrical appliances) as W =P t

(1.3)

In the above the unit of W is in joules or watt-seconds. However, because the unit cost of electrical energy is expressed in dollars per kilowatts, we express the electric power consumption in kilowatt-hour: kWh = kW × hour

(1.4)

If we let λ represent the cost of electric energy in $/kWh,then the total cost is expressed as Energy cost in dollars = kWh × λ

(1.5)

Example 1.10 Assume that you want to buy a computer. The brand A power consumption is rated as 400 W and 120 V and costs $1000; brand B power consumption is rated at 100 W and 120 V and costs $1010. Your electric company charges $0.09/kWh on your monthly bill. Compute the cost of buying the computer and the operating expense if you use your computer for 3 years at a rate of 8 hours a day. Solution At 8 hours a day for 3 years, the total operating time is given as 8 × 365 × 3 = 8760 hours. Brand A kWh energy consumption = operating time × kW of brand A = 8760 × 400 × 10 −3 = 3504 kWh

19

ESTIMATING THE COST OF ENERGY

TABLE 1.4 Fossil Fuel Emission Levels in Pounds per Billion BTU of Energy Input Pollutant Carbon dioxide (CO2)

Natural Gas

Oil

Coal

117,000

164,000

208,000

Brand B kWh energy consumption = operating time × kW of brand B = 8760 × 100 × 10 −3 = 876 kWh Total cost for brand A = brand A kWh energy consumption + cost of brand A = 3504 × 0 09 + 1000 = $1315 36 Total cost for brand B = brand B kWh energy consumption + cost of brand B = 876 × 0 09 + 1010 = $1088 84 Therefore, the total cost of operation and the price of brand B are much lower than brand A because the wattage of brand B is much less. Even though the price of brand A is lower, it is economical to buy brand B, because its operating cost is far lower than that of brand A. The carbon footprints of coal are highest among fossil fuels. Therefore, coal-fired plants produce the highest output rate of CO2 per kWh. The use of fossil fuels also adds other gases to the atmosphere per unit of heat energy as shown in Table 1.4. Example 1.11 For Example 1.7, let us assume that the electric energy is produced using coal, what is the amount of CO2 in pounds that is emitted over 3 years into the environment? What is your carbon footprint? Solution From Table 1.4, the pounds of CO2 emission per billion BTU of energy input for coal is 208,000 One kWh = 3.41 thousand BTU. Energy consumed for brand A over 3 years = 3504 × 3.41 × 103 = 11,948,640 BTU Therefore, for brand A, pounds of CO2 emitted =

11,948, 640 × 208,000 = 2485 32 lb 109

Energy consumed for brand B over 3 years = 876 × 3.41 × 103 = 2,987,160 BTU Thus, for brand B, pounds of CO2 emitted = Brand B has a much lower carbon footprint.

2, 987, 160 × 208,000 = 621 33 lb 109

20

ENERGY AND CIVILIZATION

Example 1.12 Assume that you have purchased a new high-powered computer with a gaming card and an old cathode ray tube (CRT) monitor. Assume that the power consumption is 500 W and the fuel used to generate electricity is oil. Compute the following: (i) Carbon footprints if you leave them on 24/7. (ii) Carbon footprint if it is turned on 8 hours a day. Solution (i) Hours in one year = 24 × 365 = 8760 hours Energy consumed in one year = 8760 × 500 × 10 −3 = 4380 kWh = 4380 × 3 41 × 103 = 14, 935,800 BTU From Table 1.4, pounds of CO2 emission per billion per the BTU of energy input for oil is 164,000. 14,935,800 × 164,000 Therefore, the carbon footprint for one year = 109 = 2449 47 lb 8 × footprint for 24 use in 24 hours 1 = × 2449 47 3 = 816 49 lb

(ii) Carbon footprint in the case of 8 hours day use =

1.14 NEW OIL BOOM–HYDRAULIC FRACTURING (FRACKING) The new technology in oil and gas extraction from wells deep in the Earth’s rock crust is known as hydraulic fracturing. The fracturing of the rocky crust of the Earth is accomplished by directing the pressurized mixture of water with sand and chemicals into the crust of Earth below the water lines. This technique of oil and gas extraction is known as fracking. Fracking is the method used in wells for shale gas, tight gas, tight oil, and coal seam gas and hard rock wells. It has raised environmental concerns. Water contamination, air quality, and migration of chemical to the ground surface have become a major concern of environmental groups. Fracking has reversed the decline of oil and gas production in the United States and has made the United States self-sufficient in oil and gas. Fracking methods have become a highly charged political issue. The price of the oil extracted from fracking is estimated to be around $60 per barrel (https://www.investopedia.com/articles/investing/

ESTIMATION OF FUTURE CO2

21

072215/can-fracking-survive-60-barrel.asp). Historically, the crude oil has reached an all-time high of $145.31 per barrel in July of 2008 and a record low of $1.17 per barrel in February of 1946 (https://tradingeconomics.com › Commodity). 1.15 ESTIMATION OF FUTURE CO2 The rapidly growing electrification of the developing world and their reliance on fossil fuel vehicles increased, and the demand for energy has turned developed nations to mass production of fossil fuels for power and energy. The subsequent burning of these resources has increased the amount of carbon dioxide in the Earth’s atmosphere. Using the measured proportion of CO2 in the atmosphere over the last few centuries, we can estimate the carbon dioxide concentration in future years if energy production trends are unaltered. By fitting an exponential best-fit line to the recorded data using Microsoft Excel, we can estimate the best fit as presented in Equation (1.6). With the accelerated increase in electric transportation and electric generation since the 1900s, the burning of fossil fuels has caused a rising exponential trend in the carbon dioxide concentration: Concentration = 1 853∗ e 0 0146

∗

x

+ 277

(1.6)

Here, x is the number of years since 1745 and “concentration” is the proportion of carbon dioxide in parts per million. The above equation was plotted against the available data in Figure 1.12 and projected out to the year 2100 to estimate the CO2 concentration for the year 2100 in future. The estimated carbon dioxide proportion in the year 2100, if current trends continue, is about 610 parts per million, which is nearly double the present ratio of carbon dioxide in the atmosphere. Such data can show the importance of reducing the global carbon footprint by investing in clean energy technologies and more efficient power generation. If this projection in carbon dioxide concentration would become a reality, the increased amount of CO2 in the Earth’s atmosphere would have disastrous effects. More carbon dioxide, as a significant greenhouse gas, would trap more of the sun’s energy in the Earth’s atmosphere. The increase in the greenhouse effect would increase the Earth’s average temperature. The rising temperature would speed up the melting of the polar ice caps and the rising of sea levels. Global warming will significantly change the planet’s weather patterns and ocean currents and reduce the amount of habitable land for a growing population. A dramatic increase in carbon dioxide concentration, as projected by the presented data, would have far-reaching effects on the planet’s climate, ecosystem, and atmosphere. Figure 1.12 depicts recorded and estimated annual production of CO2. The IPCC (https://www.ipcc.ch/report/ar5/wg1/) report of 2013 on climate change and global warming due to man-made carbon footprints states that

22

ENERGY AND CIVILIZATION

Measured and projected CO2 concentration in atmosphere

650 600

Proportion of CO2 (ppm)

550 500 450 400

Past data Best fit plot projection

350 300 250 1700

1750

1800

1850

1900

1950

2000

2050

2100

Year

Figure 1.12 Recorded and estimated annual production of CO2.

“the globally averaged combined land and ocean surface temperature data as calculated by a linear trend, show a warming of 0.85 [0.65 to 1.06] C, over the period 1880 to 2012, when multiple independently produced datasets exist” (http://www.climatechange2013.org/images/report/WG1AR5_SPM_ FINAL.pdf). The IPCC recommendation states that “Mitigation is a human intervention to reduce the sources or enhance the sinks of greenhouse gases.” Mitigation, together with adaptation to climate change, contributes to the objective expressed in Article 2 of the United Nations Framework Convention on Climate Change (UNFCCC): “The ultimate objective of this Convention and any related legal instruments that the Conference of the Parties may adopt is to achieve, in accordance with the relevant provisions of the Convention, stabilization of greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system.”

1.16 THE PARIS AGREEMENT | UNFCCC The main objective of Paris Agreement is as stated as follows: “Holding the increase in the global average temperature to well below 2 C above pre-industrial levels and pursuing efforts to limit the temperature increase

CONCLUSION

23

to 1.5 C above pre-industrial levels, recognizing that this would significantly reduce the risks and impacts of climate change.” The readers are encouraged to read the 29 articles of the convention that state the commitment of signing nations to keep the planet earth ecosystem for future generation (https:// unfccc.int/process-and-meetings/the-paris-agreement/the-paris-agreement). As of September 21, 2018, the US government has withdrawn from the Paris accord. 1.17 ENERGY UTILIZATION AND ECONOMIC GROWTH According to the Independent Statistics and Analysis of the US Energy Information Administration (EIA) (https://www.eia.gov/), EIA’s Annual Energy Outlook provides modeled projections of domestic energy markets through 2050. Energy production and utilization is a function of economic growth and interest rate set by the central bankers, oil price, and oil consumption around the world. EIA presents macroeconomic growth, technological progress, energy policies, and energy impact on the environment. The Zero Emission Vehicle program in California and the nine other states that chose to adopt California’s vehicle emission standard has an impact on the production of electric energy and consumption. Students are encouraged to study the EIA reports in PDF and PowerPoint presentation at https://www.eia. gov/outlooks/aeo/. The Electric Power Monthly (www.eia.gov/electricity/ monthly/) of EIA provides energy statistics from US government and up-to-date cost of various electric power assets. Also, an up-to-date research information can be obtained from the National Renewable Energy Laboratory (NREL) (https://www.nrel.gov/). NREL is dedicated to research, development, commercialization, and deployment of renewable energy and energy efficiency technologies. NREL conducts studies in all aspects of renewable energy and storage (https://www.nrel.gov/esif/labs-energy-storage.html) and learning about the capabilities, infrastructure, and research at the energy systems integration facility’s energy storage laboratory and practical issues with a solar resource and PV systems. Sustainable energy production and efficient utilization of available energy resources, thereby reducing or eliminating our carbon footprint, are some of our greatest challenges in the twenty-first century. This book addresses the problem of sustainable electric energy production as part of the design of building efficient microgrids and distributed generation and smart renewable energy grids. 1.18 CONCLUSION In this chapter, we have studied a brief history of energy sources and their utilization. The development of human civilization is the direct consequence of harnessing the Earth’s energy sources. We have used the power of the wind,

24

ENERGY AND CIVILIZATION

the sun, and wood for thousands of years. However, as new sources such as coal, oil, and gas have been discovered, we have continuously substituted a new source of energy in place of an old source. Now, we have the power to harness the solar and wind energy to make every energy user an energy producer. Global warming and environmental degradation have forced us to reexamine our energy use and resulting carbon footprints that every human must consider and be aware of its consequences. In the following chapters, we study the basic concept of power system operation, power system modeling, and the smart power grid, as well as the design of microgrids of distributed renewable energy systems.

PROBLEMS 1.1

Perform the following: (i) Write a 3000-word report summarizing the Kyoto Protocol (http:// unfccc.int/kyoto_protocol/items/2830.php). (ii) Write a 3000-word report summarizing the Paris Agreement that builds upon the Convention and for the first time brings all nations into a common cause to undertake ambitious efforts to combat climate change (The Paris Agreement | UNFCCC. https://unfccc.int/ process-and-meetings/the-paris-agreement/the-paris-agreement). (iii) Compute the simple operating margin CO2 footprint factor for a power grid load of 6000 MW if it is supplied by 50 coal-fired units with a capacity of 100 MW, 10 oil-fired generators with a capacity of 50 MW, and 10 gas-fired generators with a capacity of 50 MW or if the grid load of 6000 MW is supplied equally from gas-fired and wind- and solar-powered generators each.

1.2

Using the data given in Table 1.4, perform the following: (i) The carbon footprint of 500 W if coal is used to produce the electric power. (ii) The carbon footprint of a 500 W bulb if natural gas is used to provide the electrical power. (iii) The carbon footprint of a 500 W bulb if the wind is used to generate the electrical power. (iv) The carbon footprint of a 500 W bulb if PV energy is used to provide the electrical power.

1.3

Compute the money saved in one month by using a compact fluorescent light (CFL) bulb (18 W) instead of using an incandescent lamp (60 W) if the cost of electricity is $0.12 per kWh. Assume the lights are used for 10 hours a day.

PROBLEMS

25

1.4

Compute the carbon footprint of the lamps of problem 1 if natural gas is used as fuel to generate electricity. How much more will the carbon footprint be increased if the fuel used is coal?

1.5

Will an electric oven rated at 240 V and 1200 W provide the same heat if connected to a voltage of 120 V? If not, how much power will it consume now?

1.6

Assume the emission factor of producing electric power by PV cells is 100 g of CO2 per kWh, by wind power is 15 g of CO2 per kWh, and by coal is 1000 g of CO2 per kWh. Find the ratio of CO2 emission when (a) 15% of power comes from wind farms, (b) 5% from a PV source, and (c) the rest from coal as opposed to when all power is supplied by coalrun power stations.

1.7

Compute the operating margin of the emission factor of a power plant with three units with the following specifications over one year: Unit 1 2 3

Generation (MW)

Emission factor (lb of CO2/MWh)

160 200 210

1000 950 920

1.8

Assume the initial cost to set up a thermal power plant of 100 MW is 2 million dollars and that of a PV farm of the same capacity is 300 million dollars and the running cost of the thermal power plant is $90 per MWh and that of PV farm is $12 per MWh. Compute the time in years needed for the PV farm to become the most economical if 90% of the plant capacity is utilized in each case.

1.9

Consider a feeder that is rated 120 V and serving five light bulbs. Loads are rated 120 V and 120 W. All light loads are connected in parallel. If the feeder voltage is dropped by 20%, compute the following: (i) The power consumption by the loads on the feeder in watts. (ii) The percentage of reduction in illumination by the feeders. (iii) The amount of carbon footprint if coal is used to produce the energy.

1.10 The same as problem 1.8, except a refrigerator rated 120 V and 120 W is also connected to the feeder and voltage is dropped by 30%. (i) Compute the power consumption by the loads on the feeder in watts. (ii) Compute the percentage of reduction in illumination by the feeders. (iii) Do you expect any of the loads on the feeder to be damaged?

26

ENERGY AND CIVILIZATION

(iv) Compute the amount of carbon footprint if coal is used to produce the energy. (Hint: A 40 W incandescent light bulb produces approximately 500 lm of light.) 1.11 The same as problem 1.9, except a refrigerator rated 120 W is also connected to the feeder and voltage is raised by 30%. (i) Compute the power consumption by the loads on the feeder in watts. (ii) Compute the percentage of reduction in illumination by the feeders. (iii) Do you expect any of the loads on the feeder to be damaged? (iv) Compute the amount of carbon footprint if coal is used to produce the energy. 1.12 Compute the CO2 emission factor in pounds of CO2 per BTU for a unit in a plant that is fueled by coal, oil, and natural gas if 0.3 million tons of coal, 0.1 million barrels of oil, and 0.8 million cubic feet of gas have been consumed over one year. The average power produced over the period was 210 MW. Use the following data and the data of Table 1.4 for computation: a ton of coal has 25 million BTU, a barrel (i.e. 42 gallons) of oil has 5.6 million BTU, and a cubic foot of natural gas has 1030 BTU.

FURTHER READING Adams, V.W. (1998) The potential of fuel cells to reduce energy demands and pollution from the UK transport sector. PhD thesis, The Open University. Available at http://oro.open.ac.uk/19846/1/pdf76.pdf. Accessed October 9, 2010. BP. BP statistical review of world oil reserve. Available at https://www.bp.com/en/ global/corporate/. Accessed October 6, 2018. Durant, W. The story of civilization. Available at http://www.archive.org/details/ storyofcivilizat035369mbp. Accessed November 9, 2010. Earth Policy Institute. Climate, energy, and transportation. Available at http://www. earth-policy.org/data_center/C23. Accessed November 9, 2010. Encyclopædia Britannica. Alessandro Giuseppe Antonio Anastasio Volta. Available at http://www.britannica.com/EBchecked/topic/632433/Conte-Alessandro-Volta. Accessed November 9, 2010. Encyclopædia Britannica. Antoine-César Becquerel. Available at http://www.britannica.com/EBchecked/topic/58017/Antoine-Cesar-Becquerel. Accessed November 9, 2010. Encyclopædia Britannica. Enrico Fermi. Available at http://www.britannica.com/ EBchecked/topic/204747/Enrico-Fermi. Accessed November 9, 2010. Encyclopædia Britannica. Eugène-Melchior Péligot. Available at http://www.britannica. com/EBchecked/topic/449213/Eugene-Peligot. Accessed November 9, 2010.

FURTHER READING

27

Encyclopædia Britannica. Hans Christian Ørsted. Available at http://www.britannica. com/EBchecked/topic/433282/Hans-Christian-Orsted. Accessed November 9, 2010. Encyclopædia Britannica. James Prescott Joule. Available at http://www.britannica. com/EBchecked/topic/306625/James-Prescott-Joule. Accessed November 9, 2010. Encyclopædia Britannica. Joseph Henry. Available at http://www.britannica.com/ EBchecked/topic/261387/Joseph-Henry. Accessed November 9, 2010. Encyclopedia Britannica. Martin Heinrich Klaproth. Available at http://www.britannica.com/EBchecked/topic/319885/Martin-Heinrich-Klaproth. Accessed November 9, 2010. Encyclopædia Britannica. Michael Faraday. Available at http://www.britannica.com/ EBchecked/topic/201705/Michael-Faraday. Accessed November 9, 2010. Encyclopædia Britannica. Nikola Tesla. Available at http://www.britannica.com/ EBchecked/topic/588597/Nikola-Tesla. Accessed November 9, 2010. Energy Quest. Fossil fuels—coal, oil and natural gas (chapter 8). Available at http:// www.energyquest.ca.gov. Accessed September 26, 2010. Hertwich, E.G. and Peters, G.P. (2009) Carbon footprint of nations: a global, tradelinked analysis. Environmental Science & Technology, 43(16), 6414–6420. Available at https://pubs.acs.org/doi/full/10.1021/es803496a. Accessed October 7, 2018. Hockett, R.S. Analytical techniques for PV Si feedstock evaluation. Paper presented at the 18th workshop on Crystalline Silicon Solar Cells &Modules: Material and Processes, Vail, CO, August 2008. Intergovernmental Panel on Climate Change (IPCC). Global warming of 1.5 C. Available at http://www.ipcc.ch/report/sr15. Accessed October 7, 2018. Low Impact Life Onboard. Carbon footprints. Available at http://www.liloontheweb. org.uk/handbook/carbonfootprint. Accessed November 9, 2010. Population Reference Bureau. World population data. Available at http://www.prb. org/pdf10/10wpds_eng.pdf. Accessed November 9, 2010. Solomon, S., Qin, D., Manning, M., Alley, R.B., Berntsen, T., Bindoff, N.L., Chen, Z., Chidthaisong, A., Gregory, J.M., Hegerl, G.C., Heimann, M., Hewitson, B., Hoskins, B.J., Joos, F., Jouzel, J., Kattsov, V., Lohmann, U., Matsuno, T., Molina, M., Nicholls, N., Overpeck, J., Raga, G., Ramaswamy, V., Ren, J., Rusticucci, M., Somerville, R., Stocker, T.F., Whetton, P., Wood, R.A., and Wratt, D. (2007) Technical summary. In: Climate Change 2007: The Physical Science Basis. The Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, and H.L. Miller (eds.)). Cambridge University Press, Cambridge, UK and New York, NY. Tinazzi, M. The contribution of Francesco Zantedeschi at the development of the experimental laboratory of physics faculty of the Padua University. Available at http://www.brera.unimi.it/sisfa/atti/1999/Tinazzi.pdf. Accessed November 9, 2010. U.S. Energy Information Administration (2013). Annual Energy Outlook 2013 and Projections to 2040. Available at http://www.eia.gov/forecasts/aeo/pdf/0383(2013). pdf. Accessed December 9, 2013. U.S. Energy Information Administration. Official energy statistics from the US Government. Available at http://www.eia.doe.gov. Accessed September 26, 2010.

CHAPTER 2

POWER GRIDS 2.1

INTRODUCTION

A power grid provides electric energy to end users, who use electricity in their homes and businesses. In power grids, any device that consumes electric energy is referred to as a load. In the residential electrical systems, the loads are air conditioning, lighting, television, refrigeration, washing machine, dishwasher, etc. Similarly, the industrial loads are composite loads with induction motors forming the bulk of these loads. Commercial loads consist largely of lighting, office computers, copy machines, laser printers, communication systems, etc. All electrical loads are served at rated nominal voltages. The nominal-rated voltage of each load is specified by the manufacturer for its safe operation. In the power grid analysis, we study how to design the electric power grid network to serve the loads at their rated voltage with a maximum of 5% above or 5% below the rated nominal values. In this chapter, we introduce the basic concepts behind power grid loads: single-phase loads, three-phase loads, transformers, distribution systems, energy transmission, and modeling of the power grid using the generalized per unit (p.u.) concept. In the following chapters, we will address smart microgrids and the integration of green and renewable energy sources into interconnected bulk power grids.

Design of Smart Power Grid Renewable Energy Systems, Third Edition. Ali Keyhani. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/smartpowergrid3e

28

ELECTRIC POWER GRIDS

2.2 2.2.1

29

ELECTRIC POWER GRIDS Background

Without the planning and design of power plants, the construction of thousands of miles of transmission lines, and the control of generated power to supply the loads on a second-by-second basis, a stable and reliable electric energy system on which we have come to rely would not exist. Our modern industrialized world would not have been developed without the rapid electrification that took place around the world in the early 1900s.1 Although we recognize Thomas Edison as a tireless inventor and the designer of the first direct current (DC) generating power plant in 1882,2,3 it is Nicola Tesla to whom we owe credit for the invention and design of the power grid. Tesla developed a competing electrical system to Edison’s based on alternating current (AC),2,3 which can be transformed to high voltages and transmitted across great distances. This system is in use today. Charles Curtis designed the first steam turbine generator in Newport, Rhode Island, in 1903.3,4 However, it was not until 1917 when the first long-distance AC high-voltage (HV) transmission line was constructed and then expanded across state lines that the electric power grid became an everyday power grid network in the United States. Students are encouraged to obtain the up-to-date information from the federal laboratory (https://www.eia.gov/t) on the electric energy production5 in the United States and the National Renewable Energy Laboratory (NREL) (https://www.nrel.gov/). The NREL is dedicated to research, development, commercialization, and deployment of renewable energy and energy efficiency technologies. NREL conducts studies in all aspects of renewable energy and storage.

2.2.2

The Construction of a Power Grid System

A power system grid is a network of transmission and distribution systems for delivering electric power from suppliers to consumers. The power grids use many methods of energy generation, transmission, and distribution. Following the energy crisis of the 1970s, the federal Public Utility Regulatory Policies Act (PURPA)4,5 of 1978 aimed at improving energy efficiency and increasing the reliability of electric power supplies. PURPA required open access to the power grid network for small independent power producers (IPPs). After the deregulation of the power industry, the power generation units of many power grid companies began operating as a separate business. New power generation companies entered the power market as IPPs: IPPs generate power that is purchased by electric utilities at wholesale prices. Today, power grid generating stations are owned by IPPs, power companies, and municipalities.

30

POWER GRIDS

The end-use customers are connected to the distribution systems of power grid companies who can buy power at a retail price. Power companies are tied together by transmission lines referred to as interconnections. An interconnected network is used for power transfer between power companies. Interconnected networks are also used by power companies to support and increase the reliability of the power grid for stable operation and to reduce costs. If one company is short of power due to unforeseen events, it can buy power from its neighbors through the interconnected transmission systems. The construction of a power generating station with a high capacity, say, in the range of 500 MW, may take from 5 to 10 years. Before constructing such a power generating station, a permit must be obtained from the government. Stakeholders, the local power company, and IPPs will have to undertake an economic evaluation to determine the cost of electric energy over the life of the plan as compared with the price of power from the other producers before deciding to build the plant. Under a deregulated power industry, power grid generation and the cost of electric power are determined by supply and demand. In the United States and most countries around the world, the interconnected network power grid is deregulated and is open for all power producers. The control of an interconnected network is maintained by an independent system operator (ISO). The ISO is mainly concerned with maintaining the instantaneous balance of the power grid system load and generation to ensure that the system would remain stable. The ISO performs its function by controlling and dispatching the least costly generating units to match power generated with system loads. We will study the operation of a power grid in Chapter 4. Historically, power plants are located away from heavily populated areas. The plants are constructed where water and fuel (often supplied by coal) are available. Large-capacity power plants are constructed to take advantage of economies of scale. The power is generated in a voltage range of 11–20 kV, and then the voltage is stepped up to a higher voltage before connection to the interconnected bulk transmission network. HV transmission lines are constructed in the range of 138–765 kV. These lines are mostly overhead. However, in large cities, underground cables are also used. The lines consist of copper or aluminum. A major concern in bulk power transmission is power loss in transmission lines that is dissipated as heat due to the resistance of the conductors. The power capacity is expressed as voltage magnitude times the current magnitude. High voltages would require less current for the same amount of power and less surface area of the conductor, resulting in reduced line loss. The distribution lines are normally considered lines that are rated less than 69 kV. Bulk power transmission lines are like the interstate highway systems of the energy industry, transferring bulk power along HV lines that are interconnected at strategic locations. HV transmission lines in the range of

ELECTRIC POWER GRIDS

~800 MW

31

800 –1700 MW

Coal plant

Nuclear plant

Extra high voltage 275 – 765 kV (mostly AC, some HVDC)

~200 MW Hydroelectric plant ~30 MW

~150 MW

Sub-transmission 132 – 69 kV

Industrial power plant

Medium-sized power plant Distribution grid 120 – 3.4 kV

Up to ~150 MW

Factory Rural network

~5 MW Substations

City network

City power plant

Secondary Distribution 460 – 220 V

~2 MW

Industrial customers ~400 MW farm

Wind farm Solar farm

Figure 2.1 A power system interconnected network with high green energy penetration.6

110–132 kV are referred to as sub-transmission lines. In Figure 2.1, the sub-transmission lines supply power to factories and large industrial plants. The gas turbine power plants supply power to the sub-transmission system as shown in Figure 2.1.

32

POWER GRIDS

20 kV

132 kV

P1 + jQ1

P3 + jQ3 132 kV

1 345 kV 345 kV

3

P4 + jQ4 132 kV

4

345 kV 345 kV

345 kV 345 kV 132 kV

2

5

132 kV 20 kV

P2 + jQ2

345 kV

P5 + jQ5

Figure 2.2 A five-bus bulk power grid.

Distribution systems are designed to carry power to the feeder lines and end-use customers. The distribution transformers are connected to the HV side of the transmission or the sub-transmission system. The distribution voltages are in the range of 120, 208, 240, 277, and 480 V. The service voltage of distribution systems depends on the size of the loads. The higher commercial loads are served at 480 V and higher voltages. Figure 2.2 depicts a five-bus power system. In Figure 2.2, we have two generators that are connected to the bus (node) 1 and 2. The step-up transformers that connect the generators to the bulk power network are not shown. Load centers are represented at the bus (node) 3, 4, and 5. We use the term bus because these points of interconnections are copper bars that connect elements, that is, the generator, loads, lines, etc., of the power grid. All buses (nodes) are located in a substation. In Figure 2.2, we see a generator and a line connected to bus 1. Power generators are designed to produce a three-phase AC. The three sinusoidal distributed windings or coils carry the same current. Figure 2.3 depicts the three-phase generator voltage waveforms. Power systems around the world each operate at a fixed frequency of 50 or 60 cycles per second. Based on a universal color-code convention,6 black is used for one phase of the three-phase system; it denotes the ground as the reference phase with the zero-degree angle. Red is used for the second phase, which is 120 out of phase with respect to the black phase. Blue is used for the third phase, which is also 120 out of phase with the black phase. Figure 2.3 depicts this color representation as given in each phase.7

BASIC CONCEPTS OF POWER GRIDS

1

Black

0.8

Red

33

Blue

0.6 Volts in p.u.

0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 0

50

100

150

200

250

300

350

Degrees

Figure 2.3 Three-phase generator voltage waveforms.

2.3 2.3.1

BASIC CONCEPTS OF POWER GRIDS Common Terms

First, let us define a few common terms: • The highest level of current that a conductor can carry defines its ampacity: this value is a function of the cross-sectional area of the conductor. • Power capacity of an element of the power grid is rated in volt-amps (VA). • One thousand VA is one kilovolt amp (kVA). • One thousand kVA is one megavolt amp (MVA). • Energy is the use of electric power by loads over time; it is given in kilowatt-hours (kWh). • One thousand kWh is one megawatt-hour (MWh). • One thousand MWh is one gigawatt-hours (GWh). 2.3.2

Calculating Power Consumption

The power consumption in a DC circuit can be computed as P=V I =

V2 = I2R R

(2.1)

To calculate the power consumption in a single-phase AC circuit, we need to use the complex conjugate of current and multiply it by the voltage across the load: S = V I ∗ = V I ∠ θV −θI

(2.2)

34

POWER GRIDS

In Equation (2.2), V and I are the root mean square (RMS) values of voltage and currents. The power factor (p.f.) is computed based on the phase angle between the voltage and current where the voltage is the reference phase: p f = cos θV − θI Z=

V ∠ θV I ∠ θI

(2.3)

In the above equation, θ = (θV − θI) is also the angle of impedance. The complex power has two components, namely, active power consumption, P, and reactive power consumption, Q, as shown in Equation (2.4): S = V I cos θ + jsin θ = P + jQ

(2.4)

For complex power, the voltage across the load can also be expressed as V =I Z Therefore, we also have the following expression for complex power: S = V I ∗ = I Z I ∗ = I 2Z S = V I∗ = V

V Z

∗

=

(2.5)

V2 Z∗

(2.6)

Let us review the basic inductive circuit R–L as given in Figure 2.4. In the circuit depicted in Figure 2.4, we are using the standard polarity notation. We mark the polarity of each element by a plus and a negative. These markings facilitate the application of Kirchhoff’s law of voltage. The positive terminal indicates the direction of current flow in the circuit. For example, in this circuit, the source voltage rise—from the minus terminal to the positive terminal—is equal to the drop across the resistance and inductance in the circuit.

R=5Ω + v= 120 sin ωt

+

i – + L = 0.1 H

– –

Figure 2.4 An R–L circuit.

35

BASIC CONCEPTS OF POWER GRIDS

If at t = 0, the switch is closed, the response current is given by the differential equation v = Ri + L

di dt

(2.7)

In Equation (2.7), v and i are instantaneous values of voltage and current. After some time, the current reaches steady state. Figure 2.5 depicts the responses of the voltage and current as a function of time. At the steady state, v and i can be represented by phasor forms as V and I. In this book, we are interested in the design of a power grid for steady-state operation. We can express the steady-state operation of an R–L circuit as V = Vrms ∠ θV , Vrms =

120 2

I = Irms ∠ θI , θI = ωt + θI, 0 where V is the RMS value and θV is the phase angle of V and θI is the phase angle of I (RMS) that is determined by θV and impedance angle θ(θI = θV − θ) and θI,0 is the initial angle of current: I = Irms e j

dI = jωIrms e j dt

ωt + θI, 0

ωt + θI, 0

= jωI

The above differential equation can now be presented in its steady-state forms as V = R + jωL I

(2.8)

Voltage source 100 0 –100 –200

0

0.05

0.1

0.15

Time (seconds) Current (A)

Voltage (V)

200

6 4 2 0 –2 –4

0 Transient state

0.05

0.1 Time (seconds)

Steady state

Figure 2.5 The responses of the voltage and current as a function of time.

0.15

36

POWER GRIDS

Let XL = ωL represent the inductor reactance V = R + jX L I

(2.9)

Let Z = R + jX L = Z ∠ θ,

XL >0 R

θ = tan − 1

We will have θV − θI = θ. Generally, we choose V as the reference voltage, then θV = 0: V = V ∠ 0,

I=

V = I ∠ −θ Z

Therefore, the power supplied by a power source to an inductive load can be expressed as S = VI ∗ = V I cos θ + j V I sin θ = P + jQ

(2.10)

An inductive load has a lagging p.f. Therefore, the power source supplies reactive power to the load. The reactive power supplied by the source is consumed by the inductive load. Equation (2.10) can also be expressed as S = VI ∗ = R + jX L I I ∗ = I 2 R + j I 2 XL = P + jQ Let us review the basic inductive circuit R–C as given in Figure 2.6. Figure 2.6 presets an R–C circuit supplied by an AC source. If at t = 0 the switch is closed, the system differential equation can be presented as v = RC

dvc + vc dt

(2.11)

In Equation (2.11), v and vc are instantaneous values. After some time, the transient response will die out, and the voltage across the capacitor reaches a steady state.

R=5Ω + v= 120 sin ωt

i – + VC

C = 0.005 F –

Figure 2.6 An R–C circuit supplied by an AC source.

BASIC CONCEPTS OF POWER GRIDS

37

Voltage source 200 V (V)

100 0 –100 –200

0

0.05

0.1

0.15

0.1 Steady state

0.15

Time (seconds) 30 VC (V)

20 10 0 –10 –20

0 Transient state

0.05 Time (seconds)

Figure 2.7 The voltage response (V) and capacitor voltage response (VC) as a function of time.

At steady state, the source voltage V and capacitor voltage VC can be represented by phasor V and VC. By now, you should have noticed that we are using capital letters V and I to depict RMS values. Figure 2.7 depicts the voltage response (V) and capacitor voltage response (VC) as a function of time: 120 V = Vrms ∠ θV , Vrms = 2 VC = VC, rms ∠ θVc dVC dVC = jωVC , I = C = jωCV C dt dt Therefore, we will have VC = I

1 =I jωC

−j

1 ωC

For the R–C circuit of Figure 2.7, we will have V = R− j

1 ωC

I

V = R −jX C I where XC = 1/ωC and XC is the capacitor reactance.

38

POWER GRIDS

Let Z = R− jX C = Z ∠ −θ Then, θ = tan −1

XC R

Because we choose the load V as a reference, then set θV to zero (θV = 0), and we have θV −θI = −θ V = V ∠ 0, I =

V = I ∠θ Z

Therefore, for a capacitor load, the current through the capacitor leads the voltage. The power supplied by the AC source can be expressed as S = VI ∗ = V I cos θ −j V I sin θ = P−jQ

(2.12)

The capacitive loads will supply reactive power to the source. We can also write the complex power as S = VI ∗ = R −jX C I I ∗ = I 2 R −j I 2 XC = P −jQ

(2.13)

The p.f. is computed based on the angle between the voltage and current where the voltage is the reference phase, and p. f. = cos(θV − θI) with the designation of leading or lagging. That is, for inductive loads, the load current lags the voltage, and for the capacitive load, the load current will lead the voltage. Because the impedance is the ratio of voltage over the current flowing through an impedance, the impedance of inductive loads has a positive-phase angle, and the impedance of capacitive loads has a negative-phase angle. In a renewable residential system such as photovoltaic (PV) that is designated as PV, we will have a small amount of electric energy production in the range of 5–10 kVA. Therefore, we can use a single-phase system to distribute the power to the loads. However, when we are distributing power from PV systems in the MW range, we will need to use three-phase AC systems. The three-phase AC system can be considered as three single-phase circuits. The first AC generators were single phase. However, it was recognized that the three-phase generators could produce three times as much power. However, the higher phase generators will not produce proportionally more power.8 Consider the three-phase four-wire system given in Figure 2.8.

BASIC CONCEPTS OF POWER GRIDS

Zcc

jxc Ecn +

Ean ~ –

39

~

n

ZYa

+

–

ZYc N

Zaa

jxa

– ZYb

~ Ebn + jxb

Zbb

Figure 2.8 A three-phase four-wire distribution system.

Figure 2.8 presents a three-phase four-wire distribution system. In general, three-phase systems9,10 are designed as balanced systems. We use the same structure for generators, distribution lines, and loads. Therefore, we will have the following balanced system: Xa = X b = X c = X s Zaa = Zbb = Zcc = Zline

(2.14)

ZYa = ZYb = ZYc = Zload Saa = Sbb = Scc = PL + jQL

Figure 2.9 depicts the balanced three-phase system. For balanced generators, we will have Ean = E ∠ 0 (2.15)

Ebn = E ∠ − 120 = E ∠ 240 Ecn = E ∠ 120 = E ∠ − 240 Ecn

120

120 120

Ean

Ebn

Figure 2.9 The balanced three-phase system.

40

POWER GRIDS

Zline Zgen = Rgen + jXgen Ean

Ecn +

+ ~ – Zgen = Rgen + jXgen

~ n

–

RL

RL N

Zline

– RL

~ Ebn + Zgen = Rgen + jXgen

Zline

Figure 2.10 A balanced three-phase representation.

Therefore, a balanced three-phase system consists of balanced three-phase generators, transmission lines, and loads as shown in Figure 2.10. Suppose the loads are pure resistors, that is, ZY = RL or SL = PL, and the transmission line impedance is lumped with generator impedance, Zline = Rline + jXline + Rgen + jXgen. Figure 2.10 depicts a balanced three-phase system. Then, we can write the following equations: Ean = Ia Zline + RL Ebn = Ib Zline + RL

(2.16)

Ecn = Ic Zline + RL Ia = Ib = Ic =

Ean Ean = Zline + RL Z∠θ

(2.17)

where Z = Rline + jX line + RL = R + jX = Z ∠ θ θ = tan − 1

Ia =

X R

where R = Rline + R and X = Xline

E E E ∠ − θ, Ib = ∠ − 120−θ, and Ic = ∠ 120 −θ Z Z Z

(2.18)

Therefore, for a balanced three-phase system, the currents are also balanced. We should note that the three-phase voltages are in a plane and they are out of phase by 120 from each other. Therefore, we have Ean + Ebn + Ecn = 0; or E ∠ 0 + E ∠ −120 + E ∠ 120 = 0 E + j0 + E – 0 5 – j0 866 + E – 0 5 + j0 866 = 0

(2.19)

BASIC CONCEPTS OF POWER GRIDS

Ean

Ecn + n

–

Sa

+

41

Sc

– – Ebn +

Sb

A balanced three-phase three-wire distribution system.

Figure 2.11

Similarly, the three-phase currents are in a plane and they are out of phase by 120 from each other. Therefore, we have Ia + I b + I c = I n

(2.20)

However, for a balanced three-phase load, the sum of Ia + Ib + Ic = 0; therefore In = 0. In this case, the neutral conductor does not carry any current. Thus the neutral conductor is omitted. Figure 2.11 shows a three-phase three-wire distribution system. Consider the three-phase balanced system of Figure 2.12. Let us designate the phase voltages and phase currents as shown in Figure 2.12 by Ean ,Ebn ,Ecn = line-to-neutral voltages = phase voltages Ia ,Ib , Ic = line currents or phase currents Let us write Kirchhoff’s voltage law (KVL) for a closed path around the buses a, b, and n for the three-phase system of Figure 2.13. We will have Eab = Ean – Ebn

Zline

(2.21)

Ic

Zgen = Rgen + jXgen Ecn

Ean

+

+

Zgen = Rgen + jXgen

–

n

Zline

–

ZY

ZY

~

~

Ia

N

– ~

Ebn

ZY

Eab

+ Zgen = Rgen + jXgen

Zline

Ib

Figure 2.12 A balanced three-phase system.

42

POWER GRIDS

Ecn Eca

Ebc

120

Ean

120 120

Eab Ebn

Figure 2.13 Phase voltages and line-to-line voltages.

Therefore, for line-to-line voltages, we have Eab = E ∠ 0 – E ∠ − 120 = E – E −1 – j 3 2 Eab = 3E

(2.22)

3 + j1 2 = 3E ∠ 30 V

Ebc = Ebn – Ecn = E ∠ −120 – E ∠ 120 = 3E ∠ − 90 Therefore, we will have Eab + Ebc + Eca = 0

(2.23)

Ean + Ebn + Ecn = 0

(2.24)

Let us assume that we have a balanced three-phase system with balanced loads; therefore, we will have Ia =

Ean ZY

Ib =

Ebn ZY

Ic =

Ecn ZY

(2.25)

Because the source voltages are balanced and the loads are balanced, the resulting currents are also balanced, and we will have Ia + I b + Ic = 0

(2.26)

Let us consider the balanced three-phase Δ loads as given in Figure 2.14. We will have the following phase currents (Δ currents): Eab ZΔ Ebc IBC = ZΔ Eca ICA = ZΔ

IAB =

(2.27)

43

BASIC CONCEPTS OF POWER GRIDS

Ib Ecn

Ean

Ic

Z∆

ICA C

A

n IBC

Ebn

Z∆

Z∆

IAB

B

Ia

Figure 2.14 A balanced Δ load system.

Recall that the voltages are balanced; therefore Ean = E ∠ 0 Ebn = E ∠ −120 Ecn = E ∠ 120 or Eab = 3Ean ∠ 30 = 3E ∠ 30 (2.28)

Ebc = 3Ebn ∠ 30 – 120 = 3E ∠ −90 Eca = 3Ecn ∠ 30 + 120 = 3E ∠ 150

We designate the phases as shown in Figure 2.15 as positive sequence voltages. That is, we select phase a as a reference voltage, and it is followed by phase b with 120 out of phase and phase c with 240 out of phase from

c

Ecn

abc Ean

b

a

Ebn

Figure 2.15 A balanced three-phase voltage system.

44

POWER GRIDS

phase a. The phase currents IAB, IBC, and ICA are also balanced because the loads are balanced as shown in Equation (2.29): IAB = 3E ∠ 30 ZΔ IBC = 3E ∠ −90 ZΔ

(2.29)

ICA = 3E ∠ 150 ZΔ Let us assume the following data for the system depicted in Figure 2.14: E = 10 V

ZΔ = 5 ∠ 30 Ω

Then, we will have IAB = 3 2 ∠ 0 = 3 464 ∠ 0 A IBC = 3 2 ∠ −120 = 3 464 ∠ −120 A ICA = 3 2 ∠ 120 = 3 464 ∠ 120 A The line currents can be determined by using Kirchhoff’s current law at each node of the Δ load: Ia = IAB – ICA = 3 464 ∠ 0 – 3 464 ∠ 120 = 3 3 464 ∠ − 30 Ib = IBC – IAB = 3 464 ∠ − 120 – 3 464 ∠ 0 = 3 3 464 ∠ − 150 Ic = ICA – IBC = 3 464 ∠ 120 – 3 464 ∠ −120 = 3 3 464 ∠ 90 Note that the Δ currents IAB, ICA, IBC are balanced and the line currents Ia, Ib, and Ic are also balanced. Therefore, IAB + ICA + IBC = 0 and the neutral current (Ia + Ib + Ic = 0) is always zero for a Δ-connected load. For a balanced Δ load supplied by a balanced positive sequence source, we will have Ia = 3IAB ∠ −30 Ib = 3IBC ∠ − 30 Ic = 3ICA ∠ −30 Iline = 3IΔ where line currents lag the Δ load currents by 30 . Figure 2.16 depicts the line and phase currents in a Δ-connected load. We can change a Δ-connected load to its Y-equivalent loads. Assume balanced voltages are applied to two loads so that the line currents are equal. Figure 2.17 presents Δ loads and equivalent Y-connected loads. For a Δ-connected load, we will have IA = 3IAB ∠ – 30 = 3EAB ∠ – 30 ZΔ

(2.30)

45

BASIC CONCEPTS OF POWER GRIDS

Ic

ICA

–30

IAB

Ib Ia IBC

Figure 2.16 The line and phase currents in a Δ-connected load. IA

A

IA

A Z

C

C ZY

EAB

EAB Z

ZY

N

Z

ZY B

B

Figure 2.17 Δ Loads and equivalent Y-connected loads.

For a Y-connected load, we will have IA = EAN ZY = EAB ∠ – 30 Since

3ZY ; Note EAN = EAB

3EAB ∠ – 30 ZΔ = EAB ∠ – 30

3

(2.31)

3ZY , then we will have

3ZY = ZΔ and ZY = ZΔ 3 Figure 2.18 depicts a balanced three-phase system with Y- and Δ-connected loads. We can represent the system depicted in Figure 2.18 by the three-phase system with its three-phase loads as it is represented by the one-line diagram in Figure 2.19. In a single-line diagram, the voltages are line-to-line voltage, and power consumption is specified for all three phases. We can represent the onephase equivalent circuit with a line to neutral and power consumption per phase is depicted by Figure 2.19.

46

POWER GRIDS

Zline

Zgen = Rgen + jXgen Ecn + ~ –

Ebn

n –

Z∆

+ ~ – Zgen = Rgen + jXgen

~ Ean +

Zline

Zgen = Rgen + jXgen

ZY

ZY ZY

Zline

Z∆

Z∆

Figure 2.18 A three-phase system with two loads.

ZL G

Ia

Y connected ∆ connected load load

Figure 2.19 An one-line diagram of Figure 2.18.

Figure 2.20 presents the one-phase equivalent circuit of Figure 2.18. If we closely study the three-phase systems, we recognize that the three-phase systems are three single-phase systems. Therefore, the three-phase systems can distribute three times as much as power as a single-phase system: S3ϕ = 3Sϕ = 3Vϕ Iϕ∗

(2.32)

However, phase voltage is equal to the line-to-neutral voltage, and line-toline voltage is computed as VL − L = 3VL − N = 3Vϕ

(2.33)

Therefore, the three-phase power can be expressed as S3ϕ = 3VL− L IL∗ = P3ϕ + jQ3ϕ

(2.34)

Ia Zline Ean

G

ZY

ZY

Figure 2.20 One-phase equivalent circuit of Figure 2.18.

BASIC CONCEPTS OF POWER GRIDS

47

For three-phase Y-connected systems, we will have P3ϕ = 3VL − L IL cos θ

(2.35)

Q3ϕ = 3VL− L IL sin θ

(2.36)

Also, we can write complex power as

S3ϕ =

P23ϕ + Q23ϕ

(2.37)

P3ϕ = S3ϕ cos θ

(2.38)

Q3ϕ = S3ϕ sin θ

(2.39)

and the power factor is expressed as p. f. = cos θ, lagging or leading p f = cos θ lagging or leading

(2.40)

We always need to define the p.f. with its designation of leading or lagging; when we express the p.f. as a lagging p.f., we are stating that the current of the load lags the voltage with load voltage as the reference voltage. It also follows that the p.f. can be expressed as p f = cos θ =

P leading or lagging S

(2.41)

For a lagging p.f., the reactive power, Q, is positive. Therefore, the load consumes reactive power, and the phase angle θ is positive. Similarly, for a leading p.f., the reactive power, Q, is negative. Therefore, the load generates reactive power and the load is a capacitive load, and the angle θ is negative. In the following examples, we need to use complex number operations. Refer to Appendix A on complex algebra operation. Example 2.1 Consider a three-phase 480 V, 300 kVA load with p.f. = 0.9 lagging. What is the active, reactive, and complex power of the load? Solution We have the following known data: S = 300 kVA p f = cos θ = 0 9

lagging

In this example, we like to compute P, Q from S.

48

POWER GRIDS

We know that S3ϕ and the p.f.; hence, we can compute P3ϕ: P3ϕ = S3ϕ cos θ = 300 × 0 9 = 270 kW Q3ϕ = S3ϕ sin θ = 300 × 0 4359 = 130 77 kVAr Q > 0 because p.f. is lagging: S = 270 + j130 77 = 300 ∠ cos – 1 0 9 kVA Example 2.2 Consider a three-phase 480 V, 240 kW load with p.f. = 0.8 lagging. What is the active, reactive, and complex power of the load? Solution We have the following known data: P = 240 kW p f = cos θ = 0 8 lagging We can compute Q and S from P: S = P cos θ = 240 0 8 = 300 kVA Q3ϕ = S3ϕ sin θ = 300 × 0 6 = 180 kVAr Q > 0 because p.f. is lagging: S = 270 + j180 = 300 ∠ cos – 1 0 8 kVA Example 2.3 Consider a three-phase 480 V, 180 kVA load with p.f. = 0.0 leading. What is the active, reactive, and complex power of the load? Solution We have the following data: S = 180 kVA p f = cos θ = 0 0 leading To compute P, Q from S, we can compute P3ϕ as P3ϕ = S3ϕ cosθ = 180 × 0 0 = 0 Q3ϕ = S3ϕ sin θ = 180 × −1 = −180 kVAr Q < 0 because p.f. is leading: S = 0 – j180 = 180 ∠ −90 kVAr

LOAD MODELS

2.4

49

LOAD MODELS

We can represent an inductive load by its impedance as shown in Figure 2.21. The load impedance, ZL, is an inductive load. Most power system loads are inductive. The majority of industrial, commercial, and residential motors are of the induction type. In Figure 2.21, the load voltage, VL, is a line-to-neutral voltage, and IL is the phase current supplying the load: ZL = RL + jωL = R + jX L = ZL ∠ θ R2 + XL2 , θ = tan −1

ZL =

(2.42)

XL R

The inductive load power representation is expressed by active and reactive power consumption of the load: IL =

VL ∠ 0 VL = IL ∠ − θ, IL = ∠ −θ ZL ∠ θ ZL

(2.43)

With the load voltage as the reference (i.e. VL = |VL| ∠ 0), the load current lags the voltage as shown in Figure 2.22. IL

+ VL

~

ZL

–

Figure 2.21 The inductive impedance load model.

VL ∠0 θ

IL

Figure 2.22 The load voltage and its lagging load current for inductive load depicted in Figure 2.21.

50

POWER GRIDS

Local utility

P VL

Q

IL

P + jQ

Figure 2.23 The inductive load power representation.

The complex power absorbed by a load can be expressed as SL = VL IL∗ = VL IL ∠ −θ ∗ = VL IL cos θ + j VL IL sin θ

(2.44)

SL = VL IL VA P = VL IL cos θ W Q = VL IL sin θ Vars The complex power is expressed as S = P + jQ, where θ = tan −1

Q and p f = cos θ, lagging P

An inductive load model power representation is shown in Figure 2.24. Figure 2.23 depicts inductive load consumption of both active and reactive power. Figure 2.24 depicts the capacitive impedance load model. Here, again, the load voltage is line to neutral, and the reference and load current is the phase current supplied to the load:

IL

+ ~

VL

ZL

–

Figure 2.24 The capacitive impedance load model.

LOAD MODELS

51

IL

θ

VL ∠0

Figure 2.25 The load voltage and current of a capacitive load.

ZL = R −jX c = ZL ∠ −θ ZL = IL =

R2 + Xc2 , θ = tan −1

(2.45) Xc R

VL ∠ 0 VL = IL ∠ θ, IL = ∠θ ZL ∠ −θ ZL

With the load voltage as the reference (i.e. VL = |VL| ∠ 0), the load current leads the voltage as shown in Figure 2.25. The complex power absorbed by the load is SL = VL IL∗ = VL IL ∠ θ ∗ = VL IL ∠ −θ = VL IL cos θ − j VL IL sin θ S = P− jQ, θ = tan − 1

Q P

(2.46)

and power factor p f = cos θ, leading

Therefore, the load model can be represented as Figure 2.26. As it is presented in Figure 2.26, the active power is consumed by the load, and reactive power is supplied by the capacitive load to the local power network. Recently, more variable-speed drive systems are controlled by power converters, which are controlling many types of motors. In addition, more power electronic loads have penetrated the power systems. These types of loads act as nonlinear loads and can act as both inductive and capacitive loads during their transient and steady-state operations. The p.f. corrections and voltage control and stability are active areas of research.

Local utility

P VL

Q

IL

P – jQ

Figure 2.26 The power model for a capacitive load.

52

POWER GRIDS

Vs P

VL

Q

40 kVA Vload = 220 V p.f. = 0.9 lagging

The power model for Example 2.4.

Figure 2.27

Example 2.4 For a single-phase inductive load, given in Figure 2.27, compute the line current. Solution kVA = V IL × 103 IL = 40 × 103 220 = 181 8 IL = 181 8 ∠ − 25 8 A Example 2.5 For a three-phase inductive load given in Figure 2.28, compute the line current. Solution kVA3ϕ = 2000 VL −L = 20 kV kVA = 3VL −L IL IL = 2000

3 × 20 = 57 8 A

IL = 57 8 ∠ − 25 8 P3ϕ = kW = kVA cos θ = 2000 × 0 9 = 1800 kW Q3ϕ = kVar = kVA sin θ = 2000 × sin 25 8 = 870 46 kVAr Vs VL P3� Q3�

Figure 2.28

2000 kVA Vload = 20 kV p.f. = 0.9 lagging

The power model for Example 2.5.

TRANSFORMERS IN ELECTRIC POWER GRIDS

53

V∠0

VG

Utility system

Figure 2.29 A generator operating with a lagging power factor.

VG

V ∠0

PG QG

PG3ϕ + jQG3ϕ

Figure 2.30 The equivalent circuit for Example 2.6.

VG θ

IG

Figure 2.31

The phasor relationship of VG and IG.

Example 2.6 Consider the generator in Figure 2.29. The generator is operating with a lagging power factor. Compute the active and reactive power supplied to the system. Solution Figure 2.30 depicts the equivalent circuit model of Figure 2.29, and Figure 2.31 presents the generator voltage and the generator current: ∗ S3ϕ = 3VG IG = PG3ϕ + jQG3ϕ

2.5

TRANSFORMERS IN ELECTRIC POWER GRIDS

Two facts are clear by now. The transmitted power is the product of voltage times the current. Losses in a transmission line are the square of current through the lines times the line resistance. Therefore, the transmission of a large amount of power at low voltages would have a very large power loss.

54

POWER GRIDS

For high power transmission, we need to raise the voltage and to lower the current. This problem was solved with the invention of transformers. 2.5.1

A Short History of Transformers

In the “War of Currents”3 in the late 1880s, George Westinghouse1 (1846–1914) and Thomas Edison (1847–1931) were at odds over Edison’s promotion of DC for electric power distribution and Westinghouse’s advocacy of AC, which was also Tesla’s choice.3 Because Edison’s design was based on low-voltage (LV) DC, the power losses in distribution network were too high. Lucien Gaulard of France and John Gibbs of England1 demonstrated the first AC power transformer in 1881 in London.1 This invention attracted the interest of Westinghouse, who then used Gaulard–Gibbs transformers and a Siemens AC generator in his design of an AC network in Pittsburgh. Westinghouse understood that transformers are essential for electric power transmission because they ensure acceptable power losses. By stepping up the voltage and reducing the current, power transmission losses are lessened. Ultimately, Westinghouse established that AC power was more economical for bulk power transmission and started the Westinghouse Company to manufacture AC power equipment. 2.5.2

Transmission Voltage

In early 1890, the power transmission voltage was at 3.3 kV. By 1970, power transmission voltage had reached a level of 765 kV. The standard operating voltages in the secondary distribution system are an LV range of 120–240 V for single phase and 208–600 V for three phase. The primary power distribution voltage has a range of 2.4–20 kV. The sub-transmission voltage has a range of 23 kV up to 69 kV. The HV transmission has a range of 115–765 kV. The generating voltages are in the range of 3.2–22 kV. The generating voltage is stepped up to the HV for bulk power transmission and then stepped down to sub-transmission voltage as the power system approaches the load centers of major metropolitan areas. The distribution system is used to distribute the power within the cities. The residential, commercial, and industrial loads are served at 120–600 V voltage levels.9,10 Figure 2.32 depicts a typical PV power source feeding a local distributed generation system. The PV panels are like batteries and are the source of PV power source

VDC

LV VAC

HV VAC

LV VAC

HV VAC

2 miles

DC/AC T1 P1 + jQ1

P2 + jQ2

T2 utility transformer

Local utility Pin + jQin

Figure 2.32 A photovoltaic (PV) power source feeding a radial distribution system.

TRANSFORMERS IN ELECTRIC POWER GRIDS

55

DC power. They are connected in series and parallel. The maximum operating DC voltage is 600 V. With the current DC technology of distribution, megawatts of power distribution are very expensive. If we distribute high DC power of a renewable source such as PV systems at low voltages, power losses through the distribution lines are quite high. We can step up by using DC/DC converters (i.e. boost converters). However, the costs of associated DC/DC converters and for the protection of a DC system are quite high. However, in the design of the system shown in Figure 2.32, the DC power is converted to AC using a DC/AC inverter. We will study the sizing of power converters and the design of a PV system in Chapter 3.

2.5.3

Transformers

A transformer is an element of a power grid that transfers electric power from one voltage to another voltage level through inductively coupled windings. For a single-phase transformer, each winding is wound around a single core. The two windings are magnetically coupled using the same core structure. As in an ideal transformer, the device input power is the same as its output power. This means that the input current times input voltage is equal to the product of the output voltage and output current. One of the windings is excited by an AC source. The time-varying magnetic field induces a voltage in the second winding by Faraday’s fundamental law of induction.8–13 In a transformer, the volt per turn on the secondary winding is the same as in the primary winding. We can select the ratio of turns to step up the voltage or to step it down. For HV power transmission of power, transformers are needed to step up the voltage and therefore to step down the current and reduce the transmission line power losses. Figure 2.33 depicts an ideal transformer. An ideal transformer is assumed to have zero power losses in its core and winding: N1 I1 = N2 I2 Ampere-turn

(2.47)

ϕm I1 N1

V ϕl1

N2

Load ϕl2

Figure 2.33 An ideal single-phase transformer.

56

POWER GRIDS

I1

I2

+

+

V1

V2

–

–

Figure 2.34 The schematic of an ideal transformer.

V1 V2 = Volt turn N1 N 2

(2.48)

In Figure 2.33, ϕm depicts the mutual flux linkage and ϕl1 and ϕl2 the leakage flux on either side of the transformer. The mutual flux linkage will result in the mutual inductance, and the leakage flux will result in leakage inductance. The schematic of an ideal transformer is depicted in Figure 2.34. Because the power losses in an ideal transformer are assumed zero, the input power and output power are the same: S1 = V1 I1∗ = S2 = V2 I2∗

(2.49)

A real transformer has both core and winding losses as shown in Figure 2.35. These losses are represented by R1 and R2. Here, in this representation, R1 represents the primary-side ohmic loss, and R2 denotes the secondary-side ohmic loss. It is customary to denote the side that is connected to the source as primary and the side connected to the load as secondary. For simplicity, we will number each side or call each side by their voltage levels, that is, one side of the transformer is the HV side, and the other side is the LV side: I E = Im + IC Xm = ωLm

R1 I1

Vs –

I2 IE

+ Ic Rc

(2.50)

Im jXm

R2 +

+

V1

V2

I2

Load

–

– N1

N2

Figure 2.35 The schematic of a real transformer.

57

TRANSFORMERS IN ELECTRIC POWER GRIDS

N 1 I 2 = N2 I2

(2.51)

The current IE is referred to as the excitation current. This current has two components, Im and IC. The current Im is referred to as the magnetizing current and Ic as the core current. In Equation (2.50), Lm denotes the magnetizing inductance. This inductance is computed from the mutual inductance and the inductance of each coil: Vs = Vm cos ωt Vs = Im =

Vm 2

(2.52)

∠0

Vs ∠ 0 Vs = ∠ −90 jX m Xm

(2.53)

In AC power distribution, the source voltage can be presented by Equation (2.52). The magnetizing current is computed as presented by Equation (2.53). Figure 2.36 depicts the complete transformer model. In this model, we have shown both sides of the primary transformer resistance and primary leakage reactance, and on the secondary side, we have shown the secondary-side leakage reactance and winding resistance. The core losses and magnetizing reactance are represented by the resistance RC and the reactance Xm, and they are shown on the primary side: Xp = ωe Llp , Xs = ωe Lls , and Xm = ωe Lm where Llp, Lls, and Lm are referred to as leakage inductance of primary winding, the leakage inductance of secondary winding, and the mutual inductance of the transformer, respectively. Because the transformers are designed to have very small exciting current, in the range of 2–5% of the load current, the magnetizing shunt elements of transformers are assumed to have very high impedance; therefore, the shunt elements are eliminated in the voltage

RP + jXP

Rs + jXs +

+

VP

Rc

jXm

Vs –

– NP

Figure 2.36

NS

The complete schematic of a real transformer.

58

POWER GRIDS

I 1 R1

Xl1

R2

Xl2

I2

V1

V2

N1

N2

Figure 2.37 The equivalent model of a transformer for voltage analysis.

calculation of transformers. Figure 2.38 depicts the transformer model used in voltage analysis. In Figure 2.37, the resistance, R1, denotes the winding resistance of winding number one, and Xl1 denotes the leakage reactance of the same winding. Similarly, R2 and Xl2 denote the resistance and leakage reactance of the winding number 2. In this representation, the designation of the primary winding and secondary winding are omitted because in general, either side can be connected to the load or source. Figure 2.38 depicts the equivalent transformer model of Figure 2.37, where the impedance of both windings is referred to side 1. Let us assume side 1 of a transformer connected to the HV side and the side 2 to the LV side. L side as shown in Figure 2.38. The above equivalent circuit model can be described by the following equations: VHV VLV = Volt turn NHV NLV

(2.54)

NHV IHV = NLV ILV Ampere-turn

(2.55)

The terms Rsc and Xsc are referred to the short-circuit resistance and the short-circuit reactance of the transformer, respectively. These two terms are computed from the short-circuit tests on the transformer. The short-circuit

IHV RSC

jXSC

ILV

VHV

VLV

NHV

NLV

Figure 2.38 The equivalent model of a transformer.

59

MODELING A MICROGRID SYSTEM

test is performed by shorting one side of the transformer and applying a voltage in the range of 5–10% of the rated voltage and measuring the short-circuit current. The applied voltage is adjusted such that the measured short-circuit current is equal to the rated load current. We can think of the short-circuit test as a load test because the test reflects the rated load condition: Zsc = Rsc + jX sc

(2.56)

Rsc = R1 + a2 R2

(2.57)

Xsc = Xl1 + a2 Xl2

(2.58)

where a =

2.6

VHV NHV = VLV NLV

MODELING A MICROGRID SYSTEM

As part of our objective in this chapter, we will now develop a model representing a microgrid system as shown in Figure 2.39a and b. Figure 2.39a depicts the one-line diagram of a radial distribution feeder. In Figure 2.39b, we have presented the PV system and DC/AC inverter by PV power source. We will discuss the modeling of a PV system and the inverter in later sections. To calculate the voltage at the load using the model shown in Figure 2.39b requires a number of calculations because of the many transformers involved and the need to be aware of which sides of the transformers we are analyzing. First, we need to eliminate the transformers from the above model. To do this, we will normalize the system based on a common base voltage and current. To understand this normalizing method, we present the per unit system. (a) PV system

VDC

VAC

VAC

440 VAC 2 miles

DC/AC T1 P1 + jQ1

T2 Utility transformer

P2 + jQ2

Local utility Pin + jQin

(b) + PV power source

P1 + jQ1

P2 + jQ2

– T1

T2

Pin + jQin

Figure 2.39 (a) One-line diagram of a radial distribution feeder. (b) The impedance model diagram for a radial microgrid distribution feeder.

60

POWER GRIDS

2.6.1

The Per Unit System

Before we present the concept of the per unit (p.u.) system, we introduce the concept of the rated values or nominal values.9,10 To understand the rated values, let us think of a light bulb. For example, consider a 50 W light bulb. The power and voltage ratings are stamped on the light bulb as shown in Figure 2.40. The manufacturer of the light bulb is telling us that if we apply 120 V across the light bulb, we will consume 50 W. From the above values, we can calculate the impedance of the light bulb: Rlight_bulb =

V2 120 = P 50

2

=

14, 400 = 288 Ω 50

We also know that if we apply 480 V to the light bulb, which is four times the rated value, then we will have a bright glow and possibly the light bulb will explode. Therefore, the rated values tell us the safe operating condition of an electrical device. Let us set the following values: Pbase = 50 W = Prated Vbase = 120 V = Vrated Now let us use these values and normalize the operating condition of the light bulb: Pp u =

Pactual Pbase

(2.59)

Vp u =

Vactual Vbase

(2.60)

Therefore, for our light bulb, we will have Pp u =

50 = 1 0p u W 50

50 W 120 V

Figure 2.40 The rated values of a light bulb.

MODELING A MICROGRID SYSTEM

Vp u =

61

120 = 1 0p u V 120

Therefore, one per unit represents the full load or rated load. Let us assume that we apply 480 V across the light bulb: Vp u =

480 = 4 0p u V 120

The per unit (p.u.) voltage applied to the load is 4 p.u. or 4 times the rated voltage. When we say a device is loaded at half load, the per unit load is 0.5 p.u. If we say the applied voltage is 10% above the rated voltage, we mean 1.10 p.u. V. In general, in the per unit system, the voltages, currents, powers, impedances, and other electrical quantities are expressed on a per unit basis: Quantity per unit =

Actual value Base value of quantity

(2.61)

It is customary to select two base quantities to define a given per unit system. We normally select voltage and power as base quantities. Let us assume Vb = Vrated Sb = Srated

(2.62) (2.63)

Then, base values are computed for currents and impedances: Ib =

Sb Vb

(2.64)

Zb =

Vb Vb2 = Ib Sb

(2.65)

The per unit system values are Vp u =

Vactual Vb

(2.66)

Ip u =

Iactual Ib

(2.67)

Sp u =

Sactual Sb

(2.68)

Zp u =

Zactual Zb

(2.69)

Z

= Zp u × 100

Percent of base Z

(2.70)

62

POWER GRIDS

Example 2.7 An electrical lamp is rated 120 V, 500 W. Compute the per unit and percent impedance of the lamp. Give the p.u. equivalent circuit.

Solution We compute lamp resistance from the rate power consumption and rated voltage as P=

V2 R

R=

p f =1 0

V2 120 2 = 28 8 Ω = 500 P

Z = 28 8 ∠ 0 Ω

Select base quantities as Sb = 500 VA Vb = 120 V Compute base impedance: Zb =

Vb2 120 2 = = 28 8 Ω Sb 500

The per unit impedance is Zp u =

Z 28 8 ∠ 0 = 1 ∠ 0p u = Zb 28 8

The percent impedance is Z

= 100

A per unit equivalent circuit is given in Figure 2.41.

+ VS = 1∠ 0 p.u.

Z = 1∠ 0 p.u. –

Figure 2.41 The equivalent circuit of Example 2.7.

MODELING A MICROGRID SYSTEM

63

Example 2.8 An electrical lamp is rated at 120 V, 500 W. If the voltage applied across the lamp is twice the rated value, compute the current that flows through the lamp. Use the per unit method. Solution Vb = 120 V Vp u =

V 240 ∠ 0 = 2 ∠ 0p u = Vb 120

Zp u = 1 ∠ 0 p u The per unit equivalent circuit is as follows. Figure 2.42 depicts the per unit equivalent circuit of Example 2.8: Ip u = Ib =

Vp u 2 ∠ 0 = 2 ∠ 0p u A = Zp u 1 ∠ 0 Sb 500 = 4 167 A = Vb 120

Iactual = Ip u Ib = 2 ∠ 0 × 4 167 = 8 334 ∠ 0 A The per unit system for a one-phase circuit is as follows: Sb = S1 −ϕ = Vϕ Iϕ

(2.71)

Vϕ = Vline-to-neutral

(2.72)

where

Iϕ = Iline-current

(2.73)

For transformers, we select the bases as VbLV = VϕLV

VbHV = VϕHV

(2.74)

+ VS = 2 ∠ 0 p.u.

Z = 1∠ 2 p.u. –

Figure 2.42

The per unit equivalent circuit of Example 2.8.

64

POWER GRIDS

IbLV =

Sb VbLV

IbHV =

Sb VbHV

(2.75)

The base impedances on the two sides of the transformer are VbLV VbLV = IbLV Sb

2

ZbLV =

VbHV VbHV = IbHV Sb

2

ZbHV =

(2.76) (2.77)

Sp u =

S = Vp u Ip∗ u Sb

(2.78)

Pp u =

P = Vp u Ip u cos θ Sb

(2.79)

Qp u =

Q = Vp u Ip u sin θ Sb

(2.80)

When we have two or more transformers, we need to normalize the impedance model to a common base. We will make the selections as follows: Selection 1 Sb1 = SA

Vb1 = VA

Then, Zb1 =

2 Vb1 Sb1

Zp u 1 =

ZL Zb1

Selection 2 Sb2 = SB

Vb2 = VB

Then Zb2 =

2 Vb2 ZL and Zp u 2 = Zb2 Sb2

2 Zp u 2 ZL Zb1 Zb1 Vb1 Sb2 × = = × 2 = Zp u 1 Zb2 ZL Zb2 Sb1 Vb2

Zp u 2 = Zp u 1

Vb1 Vb2

2

×

Sb2 Sb1

(2.81)

MODELING A MICROGRID SYSTEM

65

In general, the values that are given as nominal values (rated values) are referred to as “old” values and new selection as common base as “new.” Using this notation, the transformation between old and new bases can be written as Zp u , new = Zp u , old

2

Vb, old Vb, new

×

Sb, new Sb, old

(2.82)

We can use the p.u. concept and define the p.u. model of a transformer. Consider the equivalent circuit of a transformer model referred to low-voltage (LV) side and high-voltage (HV) side as shown in Figure 2.43. We make the following selection for the transformer bases: Vb1 = VLV, rated

(2.83)

Sb = Srated

(2.84)

Then, we can compute the base voltage for the new common base: Vb2 =

VHV 1 Vb1 = Vb1 a VLV

Zb1 =

2 Vb1 Sb

2 Zb1 Vb1 = 2 = Zb2 Vb2

Zb2 =

2 Vb2 Sb

2 Vb1

1 Vb1 a

2

= a2

The per unit impedances are Zp u 1 =

Rs + jX s Zb1 Rs a2

Rs + jXs

VLV

VHV

N1

N2 S

(1) Referred to LV side

VLV

+j

Xs a2

VHV

Define

a=

VLV VHV

=

N1 N2

VT (SW1– on and SW1+ off)

DC component of Van

VC < VT (SW1+ on and SW1– off)

Figure 3.4 (a) and (b) A pulse width modulation (PWM) voltage waveform of a single-phase DC/AC inverter with two switches.

As the magnitude of the sine wave is varied relative to the triangular wave, the fundamental component of Van varies in direct proportion with the relative amplitude of the triangular wave and sine wave. Figure 3.5 shows the waveform when the amplitude of the reference sine wave is reduced. The fundamental component of Van has the same frequency as that of a triangular wave. The magnitude of the fundamental of AC output voltage is directly proportional to the ratio of the peaks of VC and VT. This ratio is defined as the amplitude modulation index, Ma: Ma =

VCMax VTMax

(3.1)

SINGLE-PHASE DC/AC INVERTERS WITH TWO SWITCHES

97

(a) VT

VC

0 t

Ts =

(b)

PWM modulated output voltage

1

Fundamental component of Van

fs Van

Vidc Vidc 2

t

0 VC > VT (SW1– on and SW1+ off)

DC component of Van

VC < VT (SW1+ on and SW1– off)

Figure 3.5 (a) and (b) A pulse width modulation (PWM) voltage waveform of a single-phase DC/AC inverter with two switches when the sine wave has reduced amplitude.

The peak of the fundamental component of the output voltage is Van, 1 =

Vidc Ma 2

(3.2)

The instantaneous value of the output voltage will be Van =

Vidc Vidc + Ma sin ωe t + Harmonics 2 2

(3.3)

where ωe = 2π fe is the frequency of the sine wave in radian per second and fe is the frequency of the sine wave in hertz.

98

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

iidc +

+ Vidc 2

PV

Vidc

Von –

– + O Vidc 2 –

C+

SW1+

Vao C–

SW1–

D1+

iout + Vout –

a Van

Load

D1–

n

Digital controller

Figure 3.6 A single-phase inverter with two switches and load connected to the center tap position.

The output voltage has a DC component of value Vidc/2 and a fundamental with an amplitude of Ma Vidc/2 and harmonics. Ma varies from 0 to 1. The presence of DC components would cause any inductive load to get magnetically saturated. Hence, an inductive load cannot be connected between nodes “a” and “n.” However, we can eliminate the DC voltage in the output by connecting loads across the terminal Vao. Figure 3.6 shows the topology of the inverter having two capacitors with an available center tap. The load is connected between node “a” and the center tap point “o.” The capacitors C+ and C− are of equal capacitance. Therefore, each has a voltage Vidc/2 across it. The potential of point “o” is +Vidc/2 with respect to point “n.” Two capacitors used in this topology with the center tap point “o” make both the positive and negative voltages available. When SW1+ is on and SW1− is off, the voltage Van is equal to Vidc. Because Von is +Vidc/2, therefore, Vao(=Van − Von) is +Vidc/2. Similarly, when SW1− is on and SW1+ is off, then Van is zero and Vao is equal to −Vidc/2. If the capacitors are sufficiently large, they maintain a constant voltage of Vidc/2 across them irrespective of the load. With the center tap point available, the voltage across the load is between positive and negative values of Vidc/2. The DC component of the output voltage is zero and output voltage Vao can be considered alternating. The logic involved in switching is the same in the previous case. The resulting output voltage is shown in Figure 3.7. Figure 3.7a depicts the control voltage (VC) and the triangular wave (VT) that samples the control voltage VC. The sampling of VC at the sampling

SINGLE-PHASE DC/AC INVERTERS WITH TWO SWITCHES

99

(a)

VC

VT

0 t

Ts =

PWM modulated output voltage

1

Fundamental component of Vao

fs

(b) Vao = Van – Vidc /2

Vidc 2

t −Vidc 2

V C > VT (SW1– on and SW1+ off)

VC < VT (SW1+ on and SW1– off)

Figure 3.7 (a) and (b) The sine pulse width modulation (PWM) for a single-phase inverter with two switches for load connected to the center tap of the capacitor.

frequency of VT produces the waveform of Figure 3.7b where Vao is the output PWM wave voltage and the fundamental of the PWM voltage is shown in Figure 3.7b. The switching policy for two power switches is as follows: (i) If VC > VT, SW1+ is on, SW1− is off, and Van = Vidc/2. (ii) If VC < VT, SW1− is on, SW1+ is off, and Van = −Vidc/2. Again, the fundamental component of the output voltage can be varied by controlling the amplitude of the sine wave. Figure 3.8 gives the voltage waveforms for a reduced peak value of a sine wave.

100

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

(a)

VC

VT

0 t

Ts =

(b)

PWM modulated output voltage

1 fs

Fundamental component of Vao

Vao

= Van – Vidc /2

Vidc 2

t −Vidc 2

0 VC > VT (SW1– on and SW1+ off)

VC < VT (SW1+ on and SW1– off)

Figure 3.8 (a) and (b) The pulse width modulation (PWM) voltage waveform of a single-phase DC/AC inverter with two switches when the sine wave has reduced amplitude.

Figure 3.8a depicts the control voltage (VC) and the triangular wave (VT) that samples the control voltage VC. However, the amplitude of control voltage is half of the control voltage of Figure 3.7. The sampling of VC at the sampling frequency of VT produces the waveform of Figure 3.8b where Vao is the output PWM wave voltage and the fundamental of the PWM voltage is shown in Figure 3.8b. The output PWM voltage is also half of Figure 3.7b. The output voltage, in this case, has no DC component. The output voltage is the PWM voltage with the fundamental frequency of the reference control voltage and harmonics. The output voltage is given by

101

SINGLE-PHASE DC/AC INVERTERS WITH TWO SWITCHES

Vao =

Vidc Ma sin ωe t + Harmonics 2

(3.4)

where ωe = 2π fe is the frequency of the sine wave in radian per second and fe is the frequency of the sine wave in Hz. When 0 ≤ Ma ≤ 1, the amplitude of the fundamental varies linearly with the amplitude modulation index. When Ma is greater than one, it enters the nonlinear region, and as it is increased further, the fundamental output voltage saturates at (4/π) (Vidc/2) and does not increase with Ma. The fundamental frequency of the output voltage is the same as the frequency of the sine voltage. Thus, by adjusting the peak of the sine wave, the amplitude of the output voltage can be varied. Similarly, by changing the frequency of the sine wave, the output frequency is varied. For the quality of the voltage waveform, the output of the inverter should be as close to the sine wave as possible. The harmonic contents in the voltage should be minimized. For achieving the low harmonic distortion, the frequency of the triangular wave is increased as much as possible relative to the frequency of the sine wave. For measurement of the effect of higher switching frequency, the frequency modulation index is increased. The frequency of modulation index, Mf, is defined as Mf =

fs fe

(3.5)

where fs is the frequency of the triangular wave and fe is the frequency of the sine wave. The amount of harmonics in the output voltage is determined by the frequency modulation index. The amount of harmonics relative to the fundamental for the amplitude modulation index of 0.6 and different frequency modulation indices are tabulated in Table 3.1. The first column of Table 3.1 lists the order of the harmonics. The power frequency in the United States is 60 Hz. Therefore, the fundamental will have a frequency of 60 Hz. The third harmonic will have three times the frequency of the fundamental, which is 180 Hz. Similarly, the fifth, seventh, and ninth harmonics will have a frequency of 300, 420, and 540 Hz, respectively. The last TABLE 3.1 The Harmonic Content of the Output Voltage for a Different Mf with Ma Fixed at 0.6 Order of Harmonic 1 3 5 7 9

Mf = 3 (%)

Mf = 5 (%)

Mf = 7 (%)

Mf = 9 (%)

100 163 61 73 37

100 22 168 25 62

100 0.42 22 168 22

100 0.05 0.38 22 168

102

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

four columns of Table 3.1 give the harmonic content as a percentage of the fundamental. The order of the harmonics present in the output voltage is a direct consequence of Mf. The third harmonic content when Mf is 3 is 163% of the fundamental. Again, the fifth, seventh, and ninth harmonics are also more than 100% when Mf is 5, 7, and 9, respectively. When Mf is 9, the third and the fifth harmonics are practically absent (0.05 and 0.38%), while the seventh and ninth harmonics are considerable. As Mf is increased, the order of the harmonics, which are a high percentage of the fundamental, also increases. The load on an inverter is usually inductive, which acts as a lowpass filter. Therefore, the higher-order harmonics are easily filtered out. But the low-order harmonics are not. So, if the Mf is high, the low-order harmonic content of the voltage will be very small, and the high-order harmonics will be filtered out, giving near sinusoidal currents. Mf should be made as high as possible to reduce the harmonic content of the load current. Another consideration is the high switching frequency that increases the switching loss. Thus, a trade-off has to be reached between the switching loss, audible noise, and the harmonic distortion. Figure 3.9 shows the waveform of the single-phase inverter with two switches when the Mf is increased to 13. The frequency of the triangular wave is 780 Hz, which is 13 times the power frequency. It can be seen that the switches are turned off and on as shown in Figure 3.9 in a particular cycle. The higher switching frequency affects the efficiency of the inverter by increasing the switching losses. Figure 3.10 shows the details of the output voltage and the flow of currents with different switches on when the load is inductive (current lags behind the (a) VT VC

0 t

Ts =

1 fs

Figure 3.9 (a) and (b) The waveforms of a single-phase inverter with two switches with an Mf of 13.

SINGLE-PHASE DC/AC INVERTERS WITH TWO SWITCHES

Fundamental value of Va0

(b)

103

PWM modulated output voltage

Va0

Vidc 2 0 t Vidc − 2

VC > VT (SW1+ on and SW1– off)

VC < VT (SW1– on and SW1+ off)

Figure 3.9 (Continued)

voltage). The flow of current through the switches depends on the direction of current and which switch is on or off. When VC is greater than VT, SW1+ is turned on by a gate pulse sent to its base. However, the switch SW1+ cannot carry current in a negative direction. At this time, the diode D1+ is forward biased and conducts the current (see Figure 3.10a). Now, the potential at point “a” is that of the positive DC bus. This will reverse bias the diode D1− . This condition persists in region 1 as shown in Figure 3.10a. After a while, if VC is still greater than VT, the current becomes positive, and if SW1+ is still on, the SW1+ conducts current as shown in Figure 3.10b. At this stage, the small (a) +

Vidc C

–

iidc + Vidc 2 –

C+

+ O C– Vidc 2 –

VC D1+

SW1+

iout +

a SW1–

VT

Vout –

Vout

D1– Iout

n 1

Figure 3.10 The equivalent circuit of the single-phase inverter with two switches for various operating conditions. (a) SW1+ on, SW1− off, and D1+ conducting, Vao = +Vidc/2, iout < 0. (b) SW1+ on and conducting, SW1− off, Vao = +Vidc/2, iout > 0. (c) SW1− on, SW1+ off, and D1− conducting, Vao = −Vidc/2, iout > 0. (d) SW1− on and conducting, SW1+ off, Vao = −Vidc/2, iout < 0.

104

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

(b) iidc + Vidc 2 –

+

Vidc

C

VC C+

SW1+

VT

iout +

a

+O Vidc C– 2 –

–

D1+

Vout –

Vout

D1–

SW1–

Iout

n

2

(c) iidc

VC + Vidc 2 –

+

Vidc C

C+

VT

iout +

a

+ O Vidc C – 2 –

–

D1+

SW1+

Vout –

Vout

D1–

SW1–

Iout

n

3

(d) +

Vidc C

–

VC

iidc + Vidc 2 –

C+

+ O C– Vidc 2 –

VT

D1+

SW1+

iout + Vout –

a SW1–

Vout

D1– Iout

n

4

Figure 3.10 (Continued)

ohmic voltage drop across SW1+ will keep D1+ reverse biased and off, while D1− is reverse biased by the positive DC bus voltage at point “a.” This condition exists in region 2 as shown in Figure 3.10b. The switch SW1+ carries current in a positive direction, while the diode D1+ carries current in the negative direction. Therefore, the switch as a whole carries current in both directions. However, which component of the switch is carrying current depends on the biased pulse control signals and the direction of the current. Now, when VC becomes smaller than VT, SW1− is turned on. At this stage, if the current is still positive, the current flows through D1− . The potential at point “a” is that of the negative DC bus voltage. The diode D1+ is reverse biased by this voltage. This condition

SINGLE-PHASE DC/AC INVERTERS WITH TWO SWITCHES

105

is shown in Figure 3.10c in region 3. The last region is the region where the output current is negative, while VC is smaller than VT. Now, both voltage and current are negative. This condition is shown in Figure 3.10d in region 4. In this region, the current conducts through SW1− and the potential at point “a” is the same as the negative DC bus. The flow of current through SW1− causes a small voltage ohmic drop in which it keeps D1− reverse biased. D1+ is kept reverse biased by the negative DC bus potential at node “a.” The path of current for different directions of current is shown in Figure 3.10a–d. Example 3.1 A half-bridge single-phase inverter with a load connected to the center tap point of the capacitors is to provide 60 Hz at its AC output terminal and is supplied from a PV source with 380 VDC. Assume the switching frequency of 420 Hz (Mf is 7) and amplitude modulation Ma of 0.9. Write a MATLAB m-file code to present the waveforms of the inverter. Include the sine wave, triangular wave, and the output wave Vao. Solution The steps followed to write the program are as follows: 1. Input DC value (Vdc) and the peak values of VT(t) and VC (VTMax and VCMax, respectively) are assigned. 2. The frequencies of VT(t) and VC are assigned ( fT and fc, respectively). 3. Ts, the period of VT, is calculated as the inverse of fT. 4. “For loop” is now used to plot VT, VC, and Vo by varying k from 0 to 2/60 (two power cycles) in steps of Ts/500. 5. VT is defined by equations of a line. 6. Here, after plotting the wave for one period, it is shifted through integral periods to plot it for the entire time range. 7. VT is plotted in black color: plot (k, VT, “k”), where “k” indicates black. 8. VC is plotted in red color: plot (k, VT, “r”), where “r” indicates red (VT and VC are plotted on the same axis). 9. Value of Vo is decided by considering whether VC > VT or VC ≤ VT: If then else

VC > VT Van = Vdc/2 Van = −Vdc/2

10. Vo is plotted with offset −500 and in blue color indicated by “b”: plot (k, −500 + Van_new, ‘b’) 11. The points where VT and VC intersect are the points of discontinuity for Vo. Here the solid lines and dotted lines are plotted using for loops keeping x-axis constant and varying the y-axis and using appropriate colors and limits.

106

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

VT and VC

200

VT VC

100 0 –100 –200

Vo

–300 –400 –500 –600 –700

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time (second)

Figure 3.11 The pulse width modulation (PWM) waveforms of a single-phase with voltage switching for a half-bridge inverter from MATLAB coding.

The MATLAB code provides the following result. Figure 3.11 depicts the single-phase PWM. With single-phase inverters with two switches, two capacitors are needed to make the center tap point available. This makes the inverter bulky. Moreover, the switches are subjected to a voltage equal to the DC link voltage, but the output of the load can be a maximum of half that value. Therefore, the switches are underutilized. We can use four switches and increase the voltage across the load. This topology will be presented next.1–5

3.3 SINGLE-PHASE DC/AC INVERTERS WITH A FOUR-SWITCH BIPOLAR SWITCHING METHOD This DC/AC converter topology has two legs. Each leg has two controllable switches, and each switch can be turned on by sending a pulse to its base. The operation of each leg is as described before for the half-bridge single-phase converters. For the converter of Figure 3.12, SW1+ and SW2− and SW1− and SW2+ are switched as pairs. This topology allows the load voltage to vary between +Vidc and −Vidc, which is double the variation available in inverters with two switches. This results in higher voltage and allows more power handling capability of these inverters. When SW1+ and SW2− are on, the effective voltage across the load terminals is the difference in the voltages at nodes “a” and

SINGLE-PHASE DC/AC INVERTERS WITH A FOUR-SWITCH BIPOLAR SWITCHING METHOD

107

iidc Vidc Vidc

C+

D1+

SW1+

SW2+

D2+ iout

a

2

+

O

Vidc

C–

SW1–

D1–

SW2–

+ Vout –

b D2–

Load

2

– n

Digital controller

Figure 3.12

A single-phase DC/AC converter with four switches.

“b,” and in this case, the voltage is +Vidc. Again, when SW2+ and SW1− are on, load voltage is −Vidc. When sine PWM as shown in Figure 3.13 is applied, the peak of the fundamental output voltage is given by Vo, 1 = Vidc Ma

(3.6)

Similar logic is followed for an inverter with two switches. If VC > VT, then the switches SW1+ and SW2− are on and other switches are off, resulting in Van at Vidc and Vbn at zero and in an output voltage Vab (=Van − Vbn) of +Vidc. When VC < VT and the switches SW1− and SW2+ are on, we will have Van at zero voltage and Vbn at Vidc voltage, giving an output voltage Vab (=Van − Vbn) of −Vidc. Therefore, the output voltage varies between the +Vidc and −Vidc as shown in Figure 3.13. This modulation technique is known as bipolar sine PWM as the output voltage jumps between positive and negative values. Therefore, for a bipolar output voltage waveform, we have the following switching policy: If VC > VT, SW1+ and SW2− are on, other switches are off, and Vab = Vidc. If VC < VT, SW1− and SW2+ are on, other switches are off, and Vab = −Vidc. The sequences of switching operations are shown in Figure 3.14 with the details of the direction of flow of current and the output voltage when different combinations of switches are on. The output voltage, in this case, is given by Vao = Vidc Ma sinωe t + Harmonics

(3.7)

108

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

VC < VT (SW1– on and SW1+ off)

VT VC 1

t

0 2

VC > VT (SW1+ on and SW1– off) Van

1

+Vidc t

0 Vbn

2

+Vidc 0 PWM modulated output voltage

t

Vab = Van – Vbn +Vidc 0 Vout Fundamental

t −Vidc

Figure 3.13 Waveforms showing sine pulse width modulation (PWM) for singlephase inverter bipolar switching.

where ωe = 2π fe is the frequency of the sine wave in radian per second and fe is the frequency of the sine wave in Hz. The peak of the fundamental component is VidcMa with 0 ≤ Ma ≤ 1. When 0 ≤ Ma ≤ 1, the amplitude of the fundamental varies linearly with the amplitude modulation index. When Ma is greater than one, it enters the nonlinear region, and as it is increased further, the fundamental output voltage saturates at (4/π) Vidc and does not increase with Ma.

109

SINGLE-PHASE DC/AC INVERTERS WITH A FOUR-SWITCH BIPOLAR SWITCHING METHOD

(a)

iidc

VC > VT

+ SW1+

C+ Vidc 2 Vidc + C–

D1+

D2+ iout

a

O SW1–

Vidc

D1–

2

SW2–

b

(b)

Load + Vout –

D2–

n

–

iout < 0

iidc VC > VT

+ C+

2 Vidc + C–

D1+

SW1+

Vidc

D2+

SW2+

iout

a

O

D1–

SW1–

Vidc 2

SW2–

b

(c)

+ Vout – iout > 0

iidc

VC > VT

+ C+

SW1+

Vidc 2

+ C–

Load

D2–

n

–

Vidc

SW2+

D1+

SW2+

D2+ iout

a

O

D1–

SW1–

Vidc 2 –

SW2–

b

Load + Vout –

D2–

n

iout > 0

(d) iidc

VC < VT

+

Vidc

Vidc 2

2 –

C–

D1+

SW2+

D2+ iout

a

O

+ Vidc

SW1+

C+

SW1–

SW2–

D1– n

b

Load + Vout –

D2– iout < 0

Figure 3.14 Schematic representation of status of power switches in a single-phase full-bridge inverter. (a) SW1+ is on, D1+ is on and conducting (Van = Vidc); SW2− is on, D2− is on and conducting (Vbn = 0), Vout = Van − Vbn = Vidc, iout < 0. (b) SW1+ is on and conducting (Van = Vidc); SW2− is on and conducting (Vbn = 0), Vout = Van − Vbn = Vidc, iout > 0. (c) SW1− is on, D1− is on and conducting (Van = 0); SW2+ is on, D2+ is on and conducting (Vbn = Vidc), Vout = Van − Vbn = −Vidc, iout > 0. (d) SW1− is on and conducting (Van = 0); SW2+ is on and conducting (Vbn = Vidc), Vout = Van − Vbn = −Vidc, iout < 0.

110

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

The switching scheme given in Figure 3.14 results in an output voltage that jumps back and forth between positive and negative DC link voltage because the transition always takes place from positive value to negative value directly; this scheme is called bipolar sine PWM. The harmonic content of the output voltage in this scheme is the same as that of the inverter with two switches (see Table 3.1). The only difference is that the DC offset voltage, in this case, is zero. Like the inverter with two switches, the amplitude of the fundamental component is decided by the sine wave by varying the amplitude modulation index (Ma = VCMax/VTMax). The harmonic content can be reduced by increasing the frequency modulation index (Mf = fs/fe).1–5 3.3.1 Pulse Width Modulation with Unipolar Voltage Switching for a Single-Phase Full-Bridge Inverter In the bipolar PWM scheme, the output PWM voltage jumps between +Vidc and −Vidc. Therefore, the load is subjected to high-voltage (HV) fluctuations. The insulations on the load get subject to high stress for this reason. The unipolar PWM method allows the output voltage to jump between +Vidc and 0 or −Vidc and 0. The switching logic is the same as that for single-phase inverters with two switches, but here there are two legs switched independently with two sine waves. The upper switch of a leg is on when the VC for that leg is having a greater magnitude than VT. The same logic is followed for the other leg with its sine wave, which is 180 apart from the sine wave of the first leg. Figure 3.15 presents the voltage waveforms. The unipolar switching policy is stated as follows: If VC > VT and −VC < VT, SW1+ and SW2− are on, other switches are off, and Vab = Vidc. If VC < VT and −VC < VT, SW1− and SW2− are on, other switches are off, and Vab = 0. If VC < VT and −VC > VT, SW1− and SW2+ are on, other switches are off, and Vab = −Vidc. If VC > VT and −VC > VT, SW1+ and SW2+ are on, other switches are off, and Vab = 0. The output voltage is given by Vao = Vidc Ma sin ωe t + Harmonics

(3.8)

where ωe = 2π fe is the frequency of the sine wave in radian per second and fe is the frequency of the sine wave in Hz. Therefore, the peak of the fundamental component has a peak of VidcMa with 0 ≤ Ma ≤ 1. When 0 ≤ Ma ≤ 1, the amplitude of the fundamental varies linearly with the amplitude modulation

SINGLE-PHASE DC/AC INVERTERS WITH A FOUR-SWITCH BIPOLAR SWITCHING METHOD

V C < VT (SW1– on and SW1+ off)

VT

VC

111

–VC > VT (SW2+ on and SW2– off)

–VC

1

t

0 2

VC > VT (SW1+ on and SW1– off) Van

–VC < VT (SW2– on and SW2+ off) 1

+Vidc t

0 Vbn

2 +Vidc t

0 PWM modulated output voltage Vab = Van – Vbn

+Vidc t

0 Vout Fundamental

−Vidc

Figure 3.15 Waveforms for the unipolar switching scheme of a single-phase inverter with four switches and a frequency modulation index of 5.

index. When Ma is greater than one, it enters the nonlinear region, and as it is increased further, the fundamental output voltage saturates at (4/π) Vidc and does not increase with Ma. Similar to the other PWM schemes, here too the frequency and the magnitude of the output voltage are controlled by the reference sine wave, and the harmonic content of the output voltage is determined by the frequency modulation index.

112

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Example 3.2 Consider a PV source of 60 V. A single-phase inverter with four switches is used to convert DC to 50 Hz AC using a unipolar scheme. Assume Ma of 0.5 and Mf of 7. Develop a MATLAB program to generate the waveforms of the inverter showing the control voltage and the output voltage. Solution The plot shown in Figure 3.16 was obtained. Figure 3.16 depicts a plot of a unipolar switching scheme for a single-phase inverter with four switches. Similar to Table 3.1, Table 3.2 tabulates the harmonic content of the output voltage. Here, it tabulates the output voltage

0

Van

VT, VC and –VC

100

Vbn

–100

–200

Vo = Van – Vbn

–300

–500 0

0.005

0.01

0.015 0.02 Time (second)

0.025

0.03

0.035

Figure 3.16 A MATLAB plot of a unipolar switching scheme for a single-phase inverter with four switches.

TABLE 3.2 The Harmonic Content of the Output Voltage for a Different Mf with Ma Fixed at 0.6 for a Unipolar Switching Scheme Order of Harmonic 1 3 5 7 9

Mf = 3 (%)

Mf = 5 (%)

Mf = 7 (%)

Mf = 9 (%)

100 12 61 67 33

100 0.01 0.55 12 62

100 0.01 0.01 0.03 0.58

100 0.01 0.03 0.02 0.01

113

THREE-PHASE DC/AC INVERTERS

of the unipolar switching scheme. It can be seen from the table that for a frequency modulation index higher than 5, the lower-order harmonics are practically absent. The higher-order harmonics in the voltage are not usually a matter of concern because they are filtered out by the low-pass characteristics of the inductive loads. The unipolar switching, therefore, has the advantage of reduced harmonics and reduced voltage fluctuations in the load. The number of times a particular switch turns on and off is the same as bipolar switching.1–5

3.4

THREE-PHASE DC/AC INVERTERS

A three-phase inverter has three legs, one for each phase, as shown in Figure 3.17. Each inverter leg operates as a single-phase inverter.2,3 The output voltage of each leg is Van, Vbn, Vcn, where “n” refers to negative DC bus voltage computed from the input voltage, Vidc, and the switch positions. The inverter has three terminals, two inputs, and one output. The inverter input is DC power that can be supplied from a storage battery system, a PV power source, a fuel cell, or a green energy DC power source such as high-speed generators or variable-speed wind generators. The sine wave signal is supplied to a digital signal processor (DSP) controller to control the output AC voltage, power, and frequency. The DSP controller sends

+

D1+

SW1+

SW2+

D2+

Vidc Vidc 2

O

+

D1–

SW1–

DC Vidc input 2 –

Van a

D2–

SW2–

n

D3+

SW2+

SW2–

D3–

Vbn

Vcn

b

c

c b a

Switching signals

Measured Vdc and Idc

Measured Vac and Iac

Digital controller V

AC output

+

Ref – V measured

+

P,Q Ref.

– P,Q measured

Figure 3.17 The operation of an inverter as a three-terminal device.

114

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

a sequence of switching signals to control the six power switches to produce the desired output AC power.1–5 3.5 3.5.1

PULSE WIDTH MODULATION METHODS The Triangular Method

The triangular wave is modeled by equations of lines. A line equation is expressed as Y = mx + b

(3.9)

where b is the Y-intercept and m is the slope. In Figure 3.18, the Y-intercept is 0. The slope can be calculated as shown below: y2 −y1 VTMax −0 = x2 −x1 0 25Ts − 0

(3.10)

Thus, for 0 < x ≤ (Ts/4), we will have Y=

VTMax x 0 25 Ts

(3.11)

Here x represents time and Y is amplitude. This process can be repeated for each line segment: m=

y2 −y1 0 −VTMax Ts Ts VT then Van = Vdc else Van = −Vdc 10. Vo is plotted with offset −500 and in blue color indicated by “b”: plot (k, −500+Van_new, ‘b’); 11. The points where VT and VC intersect are the points of discontinuity for Vo. Here the solid lines and dotted lines are plotted using loops to keep the x-axis constant and by varying the y-axis and using appropriate colors and limits.

118

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

MATLAB Program VCMax=120*sqrt(2); %maximum value of VC VTMax=VCMax/0.9; %maximum value of VT Vdc=VTMax; %input dc value fC=60; %frequency of VC fT=420; %frequency of VT Ts=1/fT; %time period of VT Vo_new=0; for k= 0:Ts/500:2/60 % to plot from 0 to 0.04 sec in steps of Ts/500 plot (k,0,'k'); plot (k,−500,'k'); % the 2 time-axes are plotted at different offsets if(rem(k,Ts)=Ts/4 && rem(k,Ts)VT) %Vo is decided depending on VC>VT and value assigned in Vo_new Vo_old=Vo_new; Vo_new=Vdc; else Vo_old=Vo_new; Vo_new=−Vdc; end plot (k,−500+Vo_new,'b'); %Van is plotted in blue (‘b’) with offset of −500 hold on if(Vo_new ~= Vo_old) %The points of discontinuity in Van are found out for j = −500−Vdc:30: %dotted lines plotted VTMax plot (k,j); hold on end for j = −500−Vdc:.2: %continuous lines plotted between the upper −500+Vdc & lower lines %of Vo=Va − Vb plot (k,j); hold on end end end

Figure 3.19 depicts the results of the simulation of method 1.

PULSE WIDTH MODULATION METHODS

119

200 VT VT and VC

100

VC

0 –100 –200 –300

Vo

–400 –500 –600 –700 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time (second)

Figure 3.19 Plot of VT(t) and VC(t) and Vo for Example 3.3 using the equation of line of triangular method.

3.5.2

The Identity Method

This method uses identity mapping by assigning a number x to the same number x. If y (x) is equal to x, then if x = 1 then y = 1, x = 2 then y = 2, etc

(3.26)

The function that accomplishes this for a triangle wave is expressed as F x = sin − 1 sin x

(3.27)

Substituting in any numbers for x, it will result in F(x) = x. Thus, as the sine wave propagates, the arcsine will cause F(x) to look like a triangular wave, not a sine wave. Because MATLAB works in radians, this equation needs to be multiplied by 2/π. Then when the sine wave hits its max of π/2 and −π/2, it will become F x =A

2 sin − 1 sin x π

(3.28)

120

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Example 3.4 Repeat Example 3.3 by plotting the triangular wave using identity mapping. Solution The same procedure is followed as in the triangular method. The only difference is that VT is defined differently: VT = VTMax

2 asin sin 2π fs t π

The result obtained is the same as that obtained in the triangular method.

3.6

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

The three-phase inverters have six switches and six diodes as shown in Figure 3.20. A switch is formed by the pair SWi and Di (i = a, b, c), which can conduct current in both directions. The three-phase inverter consists of three limbs that lie between the DC links, with each limb having two switches. By turning on the upper switch, the output node (a, b, or c) acquires a voltage of the upper DC line. Conversely, when the lower switch of a limb is on, the output node of that limb attains a voltage of the lower DC line. By alternately turning on the upper and lower switch, the node voltage oscillates between the upper and lower DC line voltages: 3 Vidc 2 2

VL− L, RMS = Ma

(3.29)

Equation (3.29) can be approximated as VL − L, RMS = 0 612Vidc Ma +

SW1+

Vidc Vidc

D1+

SW2+

D1–

SW2–

D2+

SW2+

D3+

2 + Vidc 2 –

O SW1–

+ Van –

+

D2–

Vbn – a

n

SW2–

Load b

Digital controller

Figure 3.20

D3–

+ Vcn –

Three-phase inverter topology.

c – Vbc + – Vab +

c b a

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

121

where VL−L,RMS denotes the root mean square (RMS) value of the fundamental of output line voltage. Therefore, as was stated for single-phase inverters, the switching frequency is decided by the triangular wave VT and the output voltage is decided by the reference control voltage VC, that is, the modulating wave. The modulated output voltage of each leg has the same waveform as a single-phase converter. The frequency modulation index determines the harmonic content in the output voltage waveform. The standard frequency is 60 Hz for the United States and 50 Hz in most other countries. It is generally desirable that the frequency of the triangular wave be very high to make a high-frequency modulation index that will result in low harmonics in output voltage and the load currents. However, with an increase of the triangular frequency, the switching frequency increases, and the high switching frequency results in high switching losses. Several other factors affect the selection of switching frequency; it is also desirable to use high switching frequency due to the relative ease of filtering harmonic voltages. In addition, if the switching frequency is too high, the switches may fail to turn on and off properly; this may result in a short circuit of the DC link buses and damage to the switches. In a residential and commercial system, the switching frequency selected has to be out of the audible frequency range to reduce the high-pitch audible noise. Therefore, for most applications, the switching frequency is selected to be below 6 kHz or above 20 kHz. For most applications in a residential system, the switching frequency is selected below 6 kHz. To develop a simulation model for the above converter operation, we need to express the above waveforms by mathematical expressions. Let us assume that VC (phase a) can be expressed as VC a = Ma sin ωe t

(3.30)

where ωe = (2π/Te)t, Te = 1/fe, and fe is the frequency of desired controlled voltage and Ma =

VCMax VTMax

The expression for the triangular waveform is given by x1 = − 1 +

Ts Ts t for 0 ≤ t < 2 2

Expressing the unit of time t in radian and recalling fs = Mafe, the above expression can be rewritten as x1 = −1 +

2N ωe Ts ωe t for 0 ≤ ωe t < π 2

(3.31)

122

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

where N = 1, 2, 3, and so on. In this formulation, Mf can be selected as a variable of the simulation. We can rewrite Equation (3.31) as 2N π ωe t for 0 ≤ ωe < π Mf

(3.32)

2Mf π 2π ≤ ωe t < ωe t for N Mf π

(3.33)

x1 t = −1 + Similarly, x2 t = 3 −

The triangular waveform VT(t) can be modeled by Equations (3.32) and (3.33). To develop a simulation testbed, let us express VC (for phase a), VC (for phase b), and VC (for phase c) as VC a = Ma sin ωe t

(3.34)

VC b = Ma sin ωe t −

2π 3

(3.35)

VC c = Ma sin ωe t −

4π 3

(3.36)

and the triangular wave of VT(t) as ωe t π for 0 ≤ ωe t < π N

(3.37)

ωe t π 2π for ≤ ωe t < π N N

(3.38)

x1 t = −1 + 2N x2 t = 3 – 2N

Therefore, the algorithm PWM voltage generation steps are as follows: If VC a ≥ x1 t

or x2 t , then Van = Vidc

(3.39)

If VC b ≥ x1 t

or x2 t , then Vbn = Vidc

(3.40)

If VC c ≥ x1 t

or x2 t , then Vcn = Vidc

(3.41)

Similarly,

Otherwise, Van = 0; Vbn = 0; and Vcn = 0

(3.42)

The iidc can be computed as iidc = ia × SW1+ + ib × SW2+ + ic × SW3+

(3.43)

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

123

Figure 3.21 depicts a PWM operation of a three-phase converter. As discussed in the operation of a single-phase converter, one switch is on in each leg of the three-phase converter. A sequence of switching operation for one cycle of inverter operation is shown in Figure 3.22. For example, in phase “a,”

VT VC (phase a) V C (phase b) VC (phase c) 0 t

Modulated output voltage of Phase a Van

Vidc 0

t Modulated output voltage of Phase b

Vbn

Vidc t

0 Vab = Van – Vbn

Vab Fundamental voltage

0

Vidc t

Figure 3.21 A pulse width modulation (PWM) operation of a three-phase converter.

124

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Van with respect to the negative DC bus depends on Vidc and the switch status SW1+ and SW1− . Figure 3.22a depicts this operation mode. The control objective is the same as discussed for a single-phase converter, that is, the PWM seeks to control the modulated output voltage of each phase such that the magnitude and frequency of the fundamental inverter output (a) iidc + + V C

SW1+

D1+

SW2+

D2+

SW3+

D3+

idc

Vidc

2 + C–

a

c

b

O SW1–

Vidc

D1–

SW2–

D2–

SW3–

D3–

2 –

Load n

a

c

c

b

b a

(b) iidc + + V C

SW1+

D1+

D2+

SW2+

SW3+

D3+

idc

Vidc

2 + C–

a

c

b

O SW1–

Vidc

D1–

SW2–

D2–

SW3–

D3–

2 –

Load n

a

c

c

b

b a

(c) iidc + + V C

SW1+

D1+

SW2+

D2+

D3+

SW3+

idc

Vidc

2 + C–

Vidc 2 –

a

c

b

O SW1–

D1–

SW2–

D2–

SW3–

D3– Load

a

n

b

c

c b a

Figure 3.22 Operation of the switching sequence of a three-phase inverter. (a) Van, Vbn = Vidc, Vcn = 0. Ia, Ib > 0, Ic < 0, SW1+ on, SW2+ on, SW3− on. (b) Van, Vbn = Vidc, Vcn = 0. Ia > 0, Ib, Ic < 0, SW1+ on, D2+ on, SW3− on. (c) Van = 0, Vbn = Vcn = Vidc, Ia < 0, Ib, Ic > 0, SW1− on, SW2+ on, SW3+ on. (d) Van = 0, Vbn = Vcn = Vidc, Ia, Ic < 0, Ib > 0, SW1− on, SW2+ on, D3+ on. (e) Van = Vcn = Vidc, Vbn = 0, Ia, Ic > 0, Ib < 0, SW1+ on, SW2− on, SW3+ on. (f ) Van = Vcn = Vidc, Vbn = 0, Ia, Ib < 0, Ic > 0, D1+ on, SW2− on, SW3+ on.

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

125

(d) iidc + + V C

SW1+

D1+

SW2+

D2+

D3+

SW3+

idc

2

Vidc

+ C–

a

c

b

O SW1–

Vidc 2 –

D1–

D2–

SW2–

D3–

SW3–

Load a

n

c

c

b

b a

(e) iidc + + V C

SW1+

D1+

idc

Vidc

2 + C–

SW2+

a

D2+

D3+

SW3+

b

c

O SW1–

Vidc 2 –

D1–

D2–

SW2–

D3–

SW3–

Load a

n

c

c

b

b a

(f) iidc + + V C

SW1+

D1+

SW2+

D2+

D3+

SW3+

idc

Vidc

2 + C–

a

b

c

O

Vidc

D1–

SW1–

SW2–

D2–

SW3–

D3–

2 –

Load a

n

b

c

c b a

Figure 3.22 (Continued)

voltage are the same as the control voltage. The PWM samples the DC bus voltage, and the sampled voltage is the inverter output voltage. Example 3.5 Compute the minimum DC input voltage if the switching frequency is set at 5 kHz. Assume the inverter is rated at 207.6 V AC, 60 Hz, 100 kVA. Solution Vrated = 207 6 V, kVAr = 100 kVA, f = 60 Hz

126

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

The AC-side voltage of the inverter is given as Vpeak sin 2 π f t = 207 6 2 sin 2 π 60 t = 293 59sin 2 π 60 t Here, Vpeak = 293.59 V For the three-phase inverter using sine PWM, the line-to-line peak voltage Vdc is given as VL− L, peak = Ma 3 2 Therefore, Vdc =

VL− L, peak Ma 3

2

For the inverter to operate at the maximum value of Ma is 1, the minimum DC voltage is given as Vdc, Min =

VL − L, peak 2 293 59 = 339 01 V = 1 M 3 3 a, Max

2

Therefore, the minimum Vdc = 339.01 V. Example 3.6 Consider the three-phase radial PV distribution system given in Figure 3.23. Assume load number 1 is a three-phase load of 5 kW at a power factor of 0.85 leading and load number 2 is a three-phase load of 10 kW at a power factor of 0.9 lagging at a rated voltage of 110 V with a 10% voltage variation. Assume a photovoltaic (PV) source voltage is rated at 120 V DC. Compute the transformer ratings. Assume an ideal transformer. Figure 3.23 depicts a microgrid stand-alone PV system. If the amplitude modulation index, Ma, can be controlled from 0.7 to 0.9 in steps of 0.05, compute the following: (i) The transformer low-voltage-side rating (LV). (ii) The transformer high-voltage-side rating (HV).

PV module

LV VAC

VDC

HV VAC

DC/AC T1

P1 + jQ1 P2 + jQ2

Figure 3.23 A microgrid stand-alone photovoltaic system.

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

127

(iii) The PV system per unit model if the transformer ratings are selected as the base values. (iv) The per unit model of this PV system. Solution For the DC/AC inverter, VL− L, RMS =

3 2

Ma

Vdc 2

Let us vary the modulation index from 0.7 to 0.9 to find the suitable voltage. From Table 3.3, we see that the modulation index of 0.9 is most suitable. Therefore, VL −L, RMS = kVA for load 1 =

3 2

Ma

Vdc = 2

3 2

×0 9×

120 = 66 13 V 2

kW1 5 = 5 88 kVA = p f 1 0 85

p f angle = cos − 1 0 85 = 31 78 , leading kVA for load 2 = Load 2 =

kW2 10 kVA = pf2 09

p f angle = cos −1 0 9 = 25 84 , lagging Let an ideal transformer be selected. It is rated at 70/110. The kVA of the transformer should be greater than or equal to the total load it supplies. A 20 kVA has adequate capacity: Therefore, the p u value of load 1 = Sp u , 1 =

kVA1 5 88 = 0 29 ∠ 31 78 = 20 kVAbase

TABLE 3.3 Variation of AC Bus Voltage with Modulation Index Vdc = 120 V Modulation Index, Ma 0.70 0.75 0.80 0.85 0.90

AC Bus Voltage (Low-Voltage Side) (V)

AC Bus Voltage (High-Voltage Side) (V)

51.4 55.1 58.8 62.5 66.1

110 110 110 110 110

128

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Vb = 127.02 V

Vb = 110 V

Vb = 70 V

= + _

DC/AC inverter

Ideal transformer Z = 0 + j0

Sp.u.,1

Sp.u.,2

Sb = 20 kVA

Figure 3.24 A per unit model of Figure 3.23.

Therefore, the p u value of load 2 = Sp u , 2 = =

kVA2 kVAbase

11 11 = 0 556 ∠ − 25 84 20

Now, for the DC/AC inverter, The base value of the DC side is = =

DC side voltage × Base voltage of AC side AC side voltage 120 × 70 = 127 02 V 66 13

Therefore, the p u value of the DC side of inverter =

120 = 0 94 p u 127 02

Therefore, the p u value of the AC side of inverter =

66 13 = 0 94 p u 70

Figure 3.24 depicts the per unit model of Figure 3.23. Example 3.7 Consider the three-phase radial microgrid PV given in Figure 3.25. Assume a PV is providing power to two submersible pumps on a farm. Load number 1 is rated at 5 kVA at 120 V AC and load number 2 is rated

PV module

VDC

LV VAC

HV VAC

LV VAC

HV VAC

2 miles

DC/AC T1 P1 + jQ1

P2 + jQ2

T2 Utility transformer

Local utility Pin + jQin

Figure 3.25 A radial microgrid photovoltaic system operating in parallel with the local utility.

ANALYSIS OF DC/AC THREE-PHASE INVERTERS

129

at 7 kVA at 240 V AC. Assume a load voltage variation of 10% and an amplitude modulation index of 0.9. The distribution line has an impedance of 0.04 Ω and reactance of 0.8 Ω per mile. Assume a utility voltage of 460 V. Compute the following: (i) The transformer T1 rating at the LV side. (ii) The transformer T2 rating at the HV side. (iii) The PV system per unit model with the transformer T2 ratings selected as the base values and assuming the transformers have an impedance of 10% based on selected ratings. Solution The transformer T2 is selected as a base with a voltage of 240/460 V. It must supply the loads on the LV side and must have a kVA rating greater than the loads connected to it. We choose a transformer of rating 15 kVA (>5 + 7 = 12) and select this rating as the base. The base on the transmission line side is 240 V and 15 kVA. The impedance of the transmission line is equal to its length times the impedance per unit length: Zline = 2 × 0 04 + j0 8 = 0 08 + j1 6 Ω The base impedance for the transmission line is Zb, trans =

2 Vbase 2402 = = 3 84 VAbase 15 × 103

The p.u. value of the transmission line is equal to Zp u , trans =

Ztrans 0 08 + j1 6 = = 0 02 + j0 42 p u Ω 3 84 Zb, trans

The p.u. value of load 2 is given as S2 =

kVA2 7 = 0 467 p u kVA = kVAbase 15

Let us select a transformer T1 of 120/240 V, 15 kVA. The base on the LV side of T1 is 120 V: Therefore, the p u value of the AC side of inverter = The p.u. value of load 1 is given as S1 =

kVA1 5 = 0 33 p u kVA = kVAbase 15

120 =1p u V 120

130

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Vb = 120 V

Vb = 217.7 V

Vb = 240 V

Vb = 240 V

Vb = 460 V

= + –

DC/AC inverter

Zp.u. (T1)

Zp.u. (T2)

Zp.u. (trans)

Sp.u.,1

Sp.u., local utility

Sp.u.,2 Sb = 15 kVA

Figure 3.26 A per unit model of Figure 3.25.

The transformer impedances are as given: j0.10 p.u. Ω. For the DC/AC inverter,

VL, RMS =

3 2

Ma

Vdc 2

Let us select a modulation index of 0.9.

Vdc =

2 2 VL, RMS = Ma 3

The base value of the DC side is =

V base DC side =

2 2 × 120 = 217 7 V 09 3

DC side voltage × Base voltage of AC side AC side voltage

217 7 × 120 = 217 7 V 120

Therefore the p u value of the DC side of inverter =

217 7 = 1p u V 217 7

Figure 3.26 depicts the per unit model of Figure 3.25.

3.7

MICROGRID OF RENEWABLE ENERGY SYSTEMS

Figure 3.27 depicts a community microgrid system operating in parallel with the local utility system. The transformer T1 voltage is in the range of 240/600 VAC, and the transformer T2 that connects the PV system to the

MICROGRID OF RENEWABLE ENERGY SYSTEMS

PV array DC bus PV module

T1

131

AC bus

=

DC/AC converter PV module

Step-up transformer T2

=

Net metering

DC/AC converter Rooftop PV PV module

DC bus

Step-up transformer

Step-up transformer

DC/AC converter

=

=

1

=

DG

= = Buck/boost converter

3

2

DC/DC converter Community storage

Local utility

DG

DG

Local loads

Step-up transformer

Figure 3.27 One-line diagram for a community microgrid distribution system connected to a local utility power grid.

utility bus system is rated to step up the voltage to the utility local AC bus substation. Example 3.8 Consider a microgrid that is fed from a PV generating station with an AC bus voltage of 120 V as shown in Figure 3.28. The modulation index for the inverter is 0.9. 120 V

PV module

= DC/AC T1 converter 500 kVA 220 V/3.2 kV 500 kVA R = 0.02 p.u. X = 0.05 p.u.

ZL = 1.2 + j12 Ω Sload 400 kVA T2 3.2 kV/240 V p.f. = 0.8 lagging 500 kVA R = 0.03 p.u. X = 0.08 p.u.

Figure 3.28 The microgrid of Example 3.8.

132

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

Compute the following: (i) (ii) (iii) (iv)

The per unit value of line impedance. The per unit impedance of transformers T1 and T2. The per unit model of the load. The per unit equivalent impedance and give the impedance diagram. Assume the base voltage Vb of 240 V in the load circuit and Sb of 500 kVA.

Solution We select the voltage base and power base as specified. For the utility side, we will have Vb = 240 V

Sb = 500 kVA

The p.u. load is Sp u

load

=

400 ∠ cos −1 0 8 = 0 8 ∠ 36 87 p u kVA 500

The p.u. impedance of T2 will remain the same as its rated value of 0.03 + j0.08. The base impedance of the transmission line can be computed as Zb_3200 =

Vb2 3200 2 = 20 48 Ω = Sb 500 × 103

The per unit transmission line impedance can be computed as Ztrans line, p u =

Ztrans actual 1 2 + j12 = 0 058 + j0 586 p u Ω = 20 48 Zb_3200V

The p.u. impedance of T1 is at its rated value and also will remain the same as 0.02 + j0.05. For the inverter, VL, RMS = Vdc =

2 2 VL, RMS = Ma 3

The base value of the DC value is = =

3 2

Ma

Vdc 2

2 2 × 120 = 217 7 V 3 09 DC side voltage × Base voltage of AC side AC side voltage 217 17 × 120 = 217 17 V 120

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

Vb = 217.17 V

Vb = 120 V Vb = 13.2 kV

Vb = 13.2 kV

133

Vb = 230 V

= Inverter

Zp.u. (T1)

Zp.u. (trans.)

Zp.u. (T2) Sp.u.(load)

+ _ Sb = 500 kVA

Figure 3.29 The per unit model of Figure 3.28.

Therefore, the p u value of the DC side of inverter =

217 17 = 1p u 217 17

Figure 3.29 depicts the per unit model of Figure 3.28.

3.8

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

The block diagram of a DC/DC converter is depicted in Figure 3.30. The DC/DC converter is a three-terminal device. The input voltage is converted to a higher or lower output voltage as the switching frequency is controlled. Therefore, DC/DC converter is an electronic transformer similar to tapchanging transformers. The duty cycle D defines the relationships between the input voltage and output voltage. A switching signal (see Figure 3.30) provides the command to the switch of the converter, which can be used to vary the value of duty cycle D. Depending on whether the output voltage is lower or higher than the input value, the converter is called a buck or boost or buck– boost converter. The components of DC/DC converters are an inductor L, a capacitor C, a controllable semiconductor switch S, a diode D, and load resistance R. The electronic switches are the key elements for stepping up the input DC voltage to a higher level of DC voltage in the case of a boost converter and to step down the voltage for a buck converter. The exchange of energy between an inductor and a capacitor is used for designing DC/DC converters.4 The current slew rate through a power switch is limited by an inductor. The inductor

+ Vin –

Converter

+ Vo –

D Switching signal

Figure 3.30 Block diagram of a D/DC converter.

134

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

stores the magnetic energy for the next cycle of transferring the energy to the capacitor. The slew rate is defined to describe how quickly a circuit variable changes as a function of time. The high transient switching current is damped by the switching resistance due to switching losses. The stored energy is expressed in joules as a function of the current by Equation (3.44): 1 Energy stored in the inductor = L i2L 2 1 Energy stored in capacitor = C v2C 2

(3.44)

where iL is the inductor current and vC is the voltage across the capacitor. The stored energy in an inductor is recovered by a capacitor. The capacitor provides a new level of controlled output DC voltage. The fundamental voltage and current equations are diL dt

(3.45)

dvC dt

(3.46)

vL = L iC = C

The power switch is used to charge the inductor upon closing the switch. However, because the current and voltage are related by Equations (3.45) and (3.46), the energy stored in the inductor is transferred to the capacitor in the discharge phase of the switching cycle. The basic elements of the circuit topology can be reconfigured to reduce the output voltage or to increase the output voltage. 3.8.1

The Step-Up Converter

A PV module is a variable DC power source. Its output increases at the sunrise, and it has its maximum output at noon when the maximum solar energy can be captured by the solar module. The same is true for a variable-speed wind energy source. When the wind speed is low, limited mechanical energy is converted to variable frequency electrical power; in turn, the rectified variable AC power provides a LV DC power source. A DC/DC boost converter allows capturing a wider range of DC power by boosting the DC voltage. The higher DC voltage will be in a range that can be converted to AC power at a system operating frequency by DC/AC inverters. A step-up converter is called a boost converter; it consists of an inductor L, capacitor C, controllable semiconductor switch S, diode D, and load resistance R as depicted in Figure 3.31. The inductor draws energy from the source and stores it as a magnetic field when the switch S is on. When the switch is turned off, the energy is

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

VL + +

D

– L

135

+ iL

S

Vin

C

VO

R

–

–

Figure 3.31 A boost converter circuit.

transferred to the capacitor. When a steady state is reached, the output voltage will be higher than the input voltage, and the magnitude depends on the duty ratio of the switch. To explain how the boost converter operates, let us assume that the inductor is charged in the previous cycle of operation and the converter is at steady-state operation. Let us start the cycle with the power switch S open. This condition is depicted in Figure 3.32a. Because the inductor is fully charged in the previous cycle, it will continue to force its current through the diode D to the output circuit and charge the capacitor. Let us assume that in the next cycle, the power switch S is closed. Now, the diode will be reverse biased by the capacitor voltage; hence, it will act as open. The equivalent circuit is shown in Figure 3.32b. Now the voltage across the inductor is equal to the input voltage, +Vin. Therefore, by Equation (3.45), the current in the inductor starts to rise from its initial value with a slope Vin/L. The output

(a)

VL +

+

D

– L

+ iL

Vin

C

S

–

–

(b)

VL +

+ Vin –

VO

R

D

– L

+ iL S

C

R

VO –

Figure 3.32 The equivalent circuit of boost converter when the switch S is (a) open and (b) closed.

136

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

IL,max diL dt iL

=

Vin L

IL,min

Son Ton Time

Figure 3.33 Charging phase: when the switch is closed, the current ramps up through the inductor.

voltage now is supplied by the charged capacitor alone. The inductor current iL is shown in Figure 3.33. Let us look at the next cycle when the power switch S is open again. Now, the energy stored in the inductor is transferred to the capacitor. The equivalent circuit is given in Figure 3.32a. At the instant when the switch is opened, the inductor has an existing current of IL,Max, as shown in Figure 3.33. This current was flowing through the switch S before it was opened. When the switch is opened, the inductor, which acts as a current source, tries to maintain its current, but the path through the switch is no longer available. Therefore, the inductor forces its current through the diode D, and the diode starts conducting, boosting the voltage up across the capacitor. The equivalent circuit of Figure 3.32a shows that the voltage across the inductor is now the difference between input and output voltages. Because the output voltage is more than the input voltage, the inductor voltage is negative. From Equation (3.45), the inductor current now starts to decrease with a slope (Vin − Vo)/L starting from IL,Max. During steady state, the inductor current must go back to its value, which it had at the beginning of the switching cycle (IL,min) when the switching cycle ends. The current waveform when the switch S is opened is shown in Figure 3.34. Figure 3.35 shows the steady-state current and voltage waveforms of the inductor for a few cycles of operation. In steady state, the average inductor voltage is zero. Therefore, in the steady-state operation, as expected, the inductor current is constant. If the average voltage of the inductor had not been zero, the average value of the inductor current would have continued to rise or fall depending on the polarity of the inductor voltage. For this case, the inductor current would not have returned to the value it started from, when the cycle began, at the end of the switching period according to Equation (3.45).

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

137

IL,max diL dt iL

=

Vin – V0 L

IL,min

IL,min

Son

Ton

Soff

Ts

Time

Figure 3.34 Discharging phase: when the switch opens, current ramps down through the inductor.

IL,max

VL, iL

Vin

iL

VL

IL,min Time Toff

Ton TS D · TS

(1 – D) · TS

Vo – Vin

Figure 3.35 The steady-state voltage and current waveform of a boost converter.

The waveforms of a boost converter in a steady state are shown in Figure 3.35. If the initial voltage of the capacitor is zero, the inductor current slowly charges up the capacitor over several cycles. The output voltage across the capacitor rises over each cycle until a steady state is reached. Let us analyze the operation of a boost converter depicted in Figure 3.31. The variable D in Figure 3.35 is defined as a duty ratio of the switch, and iL and vL are the inductor voltage and current with direction and polarity as indicated in Figures 3.31 and 3.32. The current flows from the DC source through the inductor L when the switch S is on. Then, the current flows through the switch S and back to the source. In Figure 3.32b, the diode is reverse biased by the output voltage across the capacitor. As shown in Figure 3.35, during the time

138

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

D ∙ Ts, energy is stored in the inductor, and the voltage across the inductor has a value of Vin. When the switch is turned off, the inductor forces the current through the diode and charges the capacitor and builds the output voltage through the diode during the time (1 − D) ∙ Ts, and the voltage across the inductor is −(Vo − Vin), which is negative because of Vo > Vin. At the end of time Ts, the capacitor is fully charged, and the energy of the inductor is transferred to the capacitor. For steady-state operation, the average voltage across the inductor is zero. Equating the average voltage of the inductor to zero, that is, VL = 0, we have 1 Vin DT s − VO − Vin Ts or Vo =

1 − D Ts = 0

Vin 1− D

(3.47)

where fs is switching frequency and Ts is defined as Ts = 1/fs and Ts = Ton + Toff (see Figure 3.35) is defined as the switching period. The duty ratio is defined as D=

Ton Ton = Ts Ton + Toff

(3.48)

Again, Toff = Ts −Ton = 1 −

Ton Ts = 1 − D Ts Ts

(3.49)

We know that the value of D is less than one. Therefore, the output voltage is always more than the input voltage. With the expression of the output voltage derived, the expression of the input current can be obtained from the power balance. Because the input and output power must be balanced for a lossless system, we can calculate the output current: Vin Iin = Vo Io

(3.50)

where Iin and Io are the average input and output currents, respectively. We can compute Iin = IL = Iin =

Vo Io Vin

Io 1−D

(3.51) (3.52)

Here, IL is the average value of inductor current. As can be seen from Figure 3.31, the input current and the inductor current are the same.

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

139

The ripple in the inductor current, ΔIL, is the difference in its maximum and minimum values in the steady state. It can be derived from Equation (3.45), knowing how much time the inductor is exposed to what voltage. From Figure 3.32b, it is seen that the inductor voltage is +Vin for a time D ∙ Ts. Therefore, from Equation (3.45) and Figure 3.35, ΔIL = IL, Max −IL, Min =

Vin DT s L

(3.53)

The maximum and minimum values of the inductor current can be determined from its average values and the ripple as shown in Equation (3.67): IL, Max = IL +

ΔIL , 2

IL, Min = IL −

ΔIL 2

(3.54)

It will take several cycles to reach the steady state. Initially, the capacitor voltage will be zero, and it remains zero when the switch is first closed. The inductor current flows through the switch S, and the capacitor remains cut off from the input for time Ton (see an equivalent circuit in Figure 3.32b). The capacitor will start receiving a charge only when the switch S is open at the end of time Ton. The equivalent circuit is shown in Figure 3.32a. Using Kirchhoff’s voltage law, we have Vin = L

diL + vC dt

(3.55)

The capacitor current is given by Equation (3.46). In the circuit of Figure 3.32a, the capacitor and inductor currents are the same for no load. By putting iC = iL in Equation (3.46) and using it in Equation (3.55), we get Vin = LC

d2 vC + vC dt 2

(3.56)

Putting 1 LC = ω20 in Equation (3.56) and solving the differential equations with an initial condition, we have vC t =

ITon sin ω0 t −Ton −Vin cos ω0 t − Ton + Vin Cω0

(3.57)

At the end of the switching period, Ts, the capacitor voltage will be given by putting t = Ts in Equation (3.57). By using Toff = Ts − Ton, we get vC Ts =

ITon sin ω0 Toff − Vin cos ω0 Toff + Vin Cω0

(3.58)

Again, at the end of one switching period Ts, the switch S is the cutoff from the output side, and its voltage remains constant at the above value. In the

Capacitor voltage (V)

140

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

900 800 700 600 500 400 300 200 100 0 0

1

2

3 Time (second)

4

5

6

Figure 3.36 No-load output voltage buildup of a boost converter with a supply voltage of 40 V and a duty ratio of 50%.

next cycle, when the switch S is turned off again, the capacitor voltage starts to rise again. Ideally, at no load, the voltage across the capacitor will continue to rise without reaching the steady state. However, in practice, the resistance in the snubber circuits of the diode and the switch will cause the voltage of the capacitor to settle down at a particular value. The time taken by the capacitor voltage to reach the steady state increases as the value of the capacitance is increased. Figure 3.36 shows a rise of the capacitor voltage for a boost converter supplied by a source of 40 V with an inductance of 1 mH and a capacitance of 100 μF, operating at a 50% duty ratio. Example 3.9 Consider the microgrid of Figure 3.37. Assume the PV generating station can produce 175 kW of power, transformer T1 is rated at 5% impedance, 240/120 V, and 75 kVA, and transformer T2 is rated at 10% impedance, 240/460 V, and 150 kVA. Assume a power base of 1000 kVA and the voltage base of 460 V in transformer T2. Assume the three-phase inverter has a modulation index of 0.8 and the boost converter has a duty ratio of 0.6. Compute the following: (i) The per unit value of transformers T1 and T2 and per unit model of the load. (ii) The per unit equivalent impedance and give the impedance diagram.

AC bus DC bus PV generating station

120 V, Commercial 50 kVA loads p.f. = 0.9 lagging

DC bus DC/DC

DC/AC

Boost converter

Inverter

T1

Local utility 240 V

T2

Figure 3.37 One-line diagram of Example 3.9.

460 V

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

141

Solution The power base is selected as 1000 kVA and voltage base on the utility side is selected as 460 V. We will have Vbnew = 460 V

Sbnew = 1000 kVA

Therefore, the new impedance base on the 460 V is given as Zb460V =

Vb2 460 2 = = 0 2116 Ω Sb 1000 × 103

For the new base, the transformer T2, we will have Zp u _new = Zp u _old × Zp u _new-T2 = 0 1 ×

Vb_old Vb_new

460 460

2

×

2

Sb_new Sb_old

1000 = 0 67 p u Ω 150

The old values of the transformer T1 are given on its nameplate rating. We recalculate the new per unit impedance of T1 based on the new selected bases. For the transformer T1, we will have Zp u _new-T1 = 0 05 ×

240 240

2

×

1000 = 0 67 p u Ω 75

The base voltage on the low side of the T1 transformer is given by the ratio of the old base voltage to the new base voltage (i.e. both on the LV side and on the HV side): Vb_new LV =

Vb_new HV Vb_old LV Vb_old HV

From the transformer, T1, nameplate ratings and the new base voltage on the transmission can be computed as Vb_new LV =

240 240

120 = 120 V

The per unit load can be calculated as Sp u

load

=

50 ∠ cos − 1 0 9 = 0 05 ∠ 25 84 p u kVA 1000

142

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

For the inverter, 3

VL, RMS =

2

Ma

Vdc 2

2 2 VL, RMS = Ma 3

Vdc =

The base value of the DC side is = =

2 2 × 240 = 489 9 V 08 3

DC side voltage × Base voltage of AC side AC side voltage 489 9 × 240 = 489 9 V 240

Therefore, the p u value of the DC side of the inverter =

489 9 = 1p u V 489 9

Now, for the DC/DC converter, Vin 1− Duty ratio

Vo =

Vin = 1 −Duty ratio × Vo = 1 − 0 6 × 489 9 = 195 96 V The base value of the LV side is given as V base on DC side =

Low voltage side × Base voltage of high voltage side High voltage side

V base on DC side =

195 96 × 489 9 = 195 96 V 489 9

Therefore, the p u value of the DC side of the inverter =

195 96 =1p u V 195 96

Figure 3.38 depicts the per unit model of Figure 3.37. Vb = 195.96 V

Vb = 240 V Vb = 489.9 V = Zp.u. (T1) =

=

+ _

DC/DC converter

DC/AC inverter

Vb = 120 V Vb = 460 V

Zp.u. (T2)

Sp.u. (local

Sp.u. (commercial

unity)

load)

Sb = 1000 kVA

Figure 3.38 The per unit model of Figure 3.37.

DC/DC CONVERTERS IN GREEN ENERGY SYSTEMS

143

Steady state voltage and current plot 200

Inductor voltage (V) and current (A)

100

0

–100

–200

–300

–400 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (second)

1 –3

×10

Figure 3.39 Inductor voltage and current for Example 3.10.

Example 3.10 Write a MATLAB code to generate the voltage and current waveform of the boost converter with an input of 120 V and operating at the duty ratio of 0.75. The capacitance value used is 100 μF. Plot the inductor voltage and current waveforms in a steady state. Figure 3.39 depicts the inductor voltage and current for Example 3.10. Solution % Boost Converter clc; clear all; Vin=120; D=0.75; L=1e−3; R=20; f=5e3; T=1/f; Vo=Vin/(1−D); Io=Vo/R; Iin=Io/(1−D); I_Lavg=Iin; dI_L=Vin/L*D*T; I_Lmin=I_Lavg−dI_L/2;

%input dc value %duty ratio %inductor value %load resistance %switching frequency %switching time period %output voltage is defined

%inductor ripple current %minimum inductor current

144

MODELING OF CONVERTERS IN POWER GRID DISTRIBUTED GENERATION SYSTEMS

I_Lmax=I_Lavg+dI_L/2; V_L=0; V_Lp=Vin; V_Ln=Vin−Vo;

%maximum inductor current %positive voltage of inductor %negative voltage of inductor

for k= 0:T/500:5*T %to plot from 0 to 0.04 sec in steps of Ts/500 plot (k,0,'k'); V_L_old=V_L; if(rem(k,T) 0, that is, IL would lead V∞. This operating condition is depicted in Figure 4.40: Pmech = P3ϕ = 3

EA L − N V ∞

L−N

Xs Q3ϕ = − 3V ∞

sin δ = 3V ∞ L−N

IL sin θ

L−N

IL cos θ

(4.34) (4.35)

Now, the generator is operating with a leading power factor as shown in Figure 4.40. Example 4.5 For a 440 V generator with a synchronous reactance of 0.8 Ω, and supplying a load of 2 kW at unity power factor, find: (i) The field current if k = 0.25 V s/rad/A (Ef = k ωs If). (ii) The field current if the power factor of the load is 0.85 lagging. Assume the machine has four poles.

232

SMART POWER GRID SYSTEMS

Solution

P

(i) The phase current = IL =

3 VL cos θ

=

2000 3 × 440 × 1

= 2 62 A.

The electromagnetic field (EMF) excitation is given as EMF = Ef = Vph + Iph Xs =

440 3

∠ 0 + 2 62 ∠ 0 × 0 8 ∠ 90 = 254 04 ∠ 0 47 V

2f 2 × 60 =2 π× = 188 5 rad s P 4 Ef 254 04 = 5 39 A = The field current = If = k ωs 0 25 × 188 5 ωs = 2 π

P

(ii) The phase current = IL =

3 VL cos θ

=

2000 3 × 440 × 0 85

= 3 08 A:

The excitation EMF = Ef = Vph + Iph Xs =

440 3

∠ 0 + 3 08 ∠ −cos −1 0 85 × 0 8 ∠ 90 = 255 34 ∠ 0 47

The field current = If =

Ef 255 34 = 5 42 A = k ωs 0 25 × 188 5

The field current needs to be increased to supply lagging power factor load (overexcitation). Example 4.6 For the generator of Example 4.5, the field current is reduced to 5.093 A without disturbing the mechanical energy input to the generator. Find the power factor angle of the load. Solution Because the mechanical input remains the same, the value of active power is unaltered: The excitation EMF = Ef = k ωs If = 0 25 × 188 5 × 5 093 = 240 V P= δ = sin − 1

3 Ef

P Xs 3 Ef Vph

Vph sin δ Xs

= sin − 1

2000 × 0 8 3 × 240 × 440

3

= 0 50

Ef = Vph + Iph Xs 240 ∠ 0 5 = Iph ∠ θ =

440 3

∠ 0 + Iph ∠ θ × 0 8 ∠ 90

1 440 ∠ 0 = 17 74 ∠ 81 51 A 240 ∠ 0 5 − 0 8 ∠ 90 3

A POWER GRID STEAM GENERATOR

233

Example 4.6 illustrates that a reduction of field current from that for the unity power factor (under excitation) results in the leading power factor. We conclude that by reducing the field current, If, the supplied current would lead the voltage and we can operate the generator with the leading power factor while generating the same active power. As we can see from the above presentation, a generator of a smart grid is a three-terminal device. The mechanical power is supplied to the generator; the excitation voltage is controlled by the voltage regulator—the field current is set for the power factor of the generator. These conditions determine the active and reactive power supplied by the generator. We should also keep in mind that we can reset the power supplied by the generators during the operation because the AGC maintains the balance of loads and generation as given by Equations (4.36) and (4.37) and depicted in Figure 4.41: 6

6

PGi = i=1

PLi + Plosses

(4.36)

QLi + Qlosses

(4.37)

i=1

6

6

QGi = i=1

i=1

Let us review the fundamental objectives of the design of a smart grid system: (1) to provide quality power at a minimum price and (2) to assure the continuity of service by keeping the power grid stable. The issue of stability of the power grid is complex, and we need to study how to construct dynamic models for each element of the power grid and then to define the time duration of the study. However, we have discussed some aspects of steady-state stability through the AGC and LFC. The power system transient stability studies require more in-depth modeling of a power system by representing the dynamics of energy sources. The design of a smart grid for studying how to provide rated voltage to loads of the system requires the steady-state model of power systems using the balanced system model representation. However, so far, we

G1″

G2″

Governor Prime mover energy source Control loop

Turbine

Network

Generator Exc.

AVR

Gn″

– + V0

1

………. S1

n Sn

Figure 4.41 A generator operation as part of an interconnected power network.

234

SMART POWER GRID SYSTEMS

have studied how to develop a per unit model for transformers, loads, and generators. We will use these models to construct the system model. 4.15 POWER GRID MODELING To construct a smart power grid, we need to represent the element of the power grid in per unit system. Every element of the grid has its own voltage, power, and impedance ratings. The ratings give the users information on the safe utilization of that element. Then, we need to select a common system power base and convert all elements of the power grid to the same common base.19,20 The power grid models are needed for two fundamental engineering design problems. These design problems are power flow studies and short-circuit studies. For power flow studies, all energy sources are scheduled to inject electric power into the power grid network to satisfy the scheduled system loads. The objective of power flow studies is to compute the bus voltages. For short-circuit studies, we have already solved the power flow problem, and the bus voltages are known. The bus loads are replaced by their load impedance models. The objective of short-circuit studies is to study the “if-then condition” for calculating the fault currents and circuit breaker short-circuit current capacity for a fault anywhere in the power grid system. The scheduled generation may be supplied from conventional power plants and renewable energy sources such as wind or solar sources or green energy sources such as fuel cells, biomass, or microturbines. For power flow studies, the power sources are represented as power injection points. The internal impedance of the energy source is not used in voltage computation. There are a number of generator models that are used in power flow studies. The basic models are (1) a constant PG and QG and (2) a constant P and V model. In the constant P and Q model, the injected power into the power network is given, and the bus voltage magnitude and phase angle are computed. We will discuss the modeling of power grid network for power flow studies in Chapter 7. Figure 4.42 depicts an eight-bus power grid system. As can be seen, the transformers of PV and wind generators are grounded through a grounding

DC/AC

3

T

1 PV generating station

CB

2 CB

CB

jX

T

CB

6

jX CB

7 CB

CB

CB T

CB

5 CB

CB

Gas turbine generator unit

4 Load

Figure 4.42

An eight-bus power grid system.

DC/AC

PV generating station

235

POWER GRID MODELING

reactance. The grounding reactance function is to limit ground current if a single-line-to-ground fault happens to occur in the system and to detect ground current. The Δ–Y transformers are used to step up the voltage to the voltage level of transmission lines. The circuit breakers are located to isolate one section of the system during maintenance and unforeseen faults. The Δ–Y transformers, T1 and T2, are also providing isolation and limit the ground current flow in the system. The Δ–Y transformer, T3, provides lower voltage as the power is transferred to the loads. Besides, the transformer T3 is grounded Δ–Y to provide a grounded system on the load side. The impedance diagram for the power grid system of Figure 4.42 is shown in Figure 4.43. This impedance diagram depicts the balanced operation of the system. Because the system is balanced, the sum of the voltages at the neutral point is zero, and no ground current will flow in this system. However, if there is a single-line-to-ground fault on the system, we need to model the unbalanced system. The modeling of the unbalanced system will be addressed in Chapter 8. For short-circuit studies, we use the bus voltage and the bus load active and reactive power consumption to calculate the load current. The load impedance model is given by the ratio of bus voltage and bus load current shown in Equation (4.38): S = V I∗ I= Zload =

2

1 CB

CB

CB

V I

(4.38)

3

jX2–3 CB

∗

S V

CB

6 CB

jXt1

CB

jX3–6

7 CB

CB

jXt2 CB

CB CB

jxg2

jxg1

jXt4 CB

CB

4

5

PV1

~ Figure 4.43

jXt3

Gas turbine

Zload

The impedance diagram of Figure 4.42.

PV2

236

SMART POWER GRID SYSTEMS 3

PV station

DC/AC

T1

4

CB

6

1 + j10 CB

7

j20

CB

1

T2

CB

AC PV bus S6

S7

Local utility

0.3

j5

+ j6

+ 0.5

S4

5 S5 2

Figure 4.44

T3

Gas turbine

One-line diagram for Example 4.7.

The impedance diagram of Figure 4.43 cannot be used for power flow studies because the bus voltages are unknown, and the load impedance cannot be computed. However, for short-circuit studies, we are interested in the computation of fault currents. The internal impedance of generators, motors, and loads should be included in the power grid model because the internal impedance limits the fault current flow. For power flow studies, we need to represent the generators and load as injection power models. Example 4.7 Consider the microgrid given in Figure 4.44. The impedance of the transmission line is given in Figure 4.44. The system data are as follows: PV generating station: 2 MW, 460 V DC, 7% reactance. Gas turbine generating station: 10 MVA, 3.2 kV, 10% reactance. Transformer T1: 10 MVA, 460 V/13.2 kV, 7% reactance. Transformer T2: 25 MVA, 13.2 kV/69 kV, 9% reactance. Transformer T3: 20 MVA, 13.2 kV/3.2 kV, 8% reactance. Load S4: 4 MW at 0.9 lagging power factor. Load S5: 8 MW at 0.9 lagging power factor. Load S6: 10 MW at 0.9 leading power factor. Load S7: 5 MW at 0.85 lagging power factor. Assume the local power internal reactance is equal to 10 Ω with negligible internal resistance. Select the transformer T2 rating as a base. Perform the following: (i) The per unit impedance model for power flow studies. (ii) The per unit impedance model for short-circuit studies. Solution Selecting the rating of transformer T2 as a base, the power base is Sb = 25 MVA

POWER GRID MODELING

237

The voltage base on the PV generator side Vb = 460 V. Therefore, the voltage base on the transmission line side is Vb = 13.2 kV. The base impedance is given by Zb =

Vb2 13 22 = 6 97 Ω = 25 Sb

(i) The per unit impedance at a new base is given by Zp u , new = Zp u , old ×

Sb, new Vb, old × Sb, old Vb, new

2

Hence, the new per unit impedance of transformer T1 is × z3 – 4 = jX p u = j0 07 × 25 10

13 2 13 2

2

= j0 175 p u Ω

Because the ratings of the transformer T2 is chosen as a base, its p.u. impedance remains at

z1 – 7 = j0 09 p u Ω Hence, the new p.u. impedance of transformer T3 is × z2 – 5 = jX p u = j0 08 × 25 20

13 2 13 2

2

= j0 1 p u Ω

Table 4.3 gives the values of per unit impedances of the transformers. The per unit values of the line impedance are given by Zp u =

Z Zb

Table 4.4 gives the values of per unit impedances of the transmission line. TABLE 4.3 Per Unit Impedance of the Transformers Transformer 1 2 3

Impedance

z z z

3–4 1–7 2–5

Per Unit Impedance j0.175 j0.09 j0.1

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SMART POWER GRID SYSTEMS

TABLE 4.4 Per Unit Impedance of the Transmission Lines Line

Impedance (Ω)

4–5 4–6 5–6

z z z

Per Unit Impedance

= 0.5 + j5 4–6 = 1 + j10 5–6 = 0.3 + j6

0.072 + j0.717 0.143 + j1.434 0.043 + j0.861

4–5

TABLE 4.5 Per Unit Loads Bus 4 5 6 7

Load (MVA)

Per Unit Load

S4 = −4 − j1.94 S5 = −8 − j3.87 S6 = −10 + j4.84 S7 = −5 − j3.10

−0.16 − j0.078 −0.32 − j0.155 −0.40 + j0.194 −0.20 − j0.124

6

4

3

1

7 6–7

4–6

3–4

1–7

S3

S1 AC PV bus

S4

S6 4–5

S7

Local utility

5–6

5 S5

2–5

Gas turbine

2 S2

Figure 4.45 Model for power flow studies.

The per unit loads of the system are given by Sp u =

S Sb

Table 4.5 gives the values of per unit loads. Figure 4.45 gives the per unit model of the system of Example 4.7 for power flow studies. (ii) For short-circuit studies, the internal impedances of the generators are also included in the model. The p.u. impedance of PV station at the new base is

z3 – 3 = jX p u = j0 07 × 252 ×

13 2 13 2

2

= j0 875 p u Ω

POWER GRID MODELING

239

The new p.u. impedance of the gas turbine generator is × z2 – 2 = jX p u = j0 1 × 25 10

13 2 13 2

2

= j0 25 p u Ω

The new p.u. impedance of the local power grid is Zp u =

Z = Zb

j10 692 25

= j0 052

z1 – 1 = j0 052 p u Ω Table 4.6 gives the values of p.u. internal impedances of the generators. The p.u. model for short-circuit studies is given in Figure 4.46. The p.u. impedances of transformers and the transmission lines are given in Tables 4.3 and 4.4, respectively.

TABLE 4.6 Per Unit Values of the Internal Impedance of the Generators Generator

Impedance

Local power grid Gas turbine PV

z z z

3

3ʹ 3ʹ–3

Per Unit Impedance j1.43 j0.25 j0.875

1 –1 2 –2 3 –3

4

6

3–4

4–6

1

7 6–7

1–7

1ʹ 1ʹ–1

Local utility

PV generating station

4–5

5–6

5 2–5

2

2ʹ–2 2ʹ

Gas turbine

Figure 4.46 Model for short-circuit studies.

240

SMART POWER GRID SYSTEMS

In this chapter, we have learned about the fundamental operation of a power grid and how to model the power grid for the analysis and design of a smart grid. We have also presented the important elements of a smart grid and load dynamics including how load variation during daily operation affects the price of electric energy. The importance of generator and motor internal impedance for limiting fault current power grid network was emphasized. Finally, the problems of power flow studies and short-circuit studies for the design of a power grid were presented. In the following chapter, we will study the design of smart MRG.

PROBLEMS 4.1

A three-phase generator rated at 440 V, 20 kVA is connected through a cable with an impedance of 4 + j15 Ω to two loads: (i) A three-phase, Y-connected motor load, rated at 440 V, 8 kVA, p.f. of 0.9 lagging. (ii) A three-phase, Δ-connected motor load, rated at 440 V, 6 kVA, p.f. of 0.85 lagging. If the motor load voltage is to be 440 V, find the required generator voltage.

4.2

A generator is rated at 100 MVA, 20 kV, 60 Hz, 0.8 p.f. lagging and reactance of 10%. Compute the following: (i) The generator per unit model if it is loaded at 50%. (ii) The generator per unit model if it is loaded at 100%. (iii) The number of poles in the generators if the shaft power is supplied at 1200 rpm.

4.3

Develop a table showing the speed of magnetic field rotation in AC machines with two, four, and six poles operating at frequencies 50, 60, and 400 Hz.

4.4

A 20 MVA machine rated 20 kV, three-phase, 60 Hz generator is supplying power to the local power grid at rated machine voltage. The machine is delivering the rated power to the local power grid. The machine synchronous reactance is equal to 8 Ω with negligible resistance. Compute the following: (i) The machine excitation voltage when the machine is operating at 0.85 lagging power factor. (ii) The machine excitation voltage when the machine is operating at 0.85 leading power factor. (iii) The machine excitation voltage when the machine is operating at unity power factor. (iv) The MW of power the machine can deliver for (i), (ii), and (iii).

PROBLEMS

241

4.5

For Problem 4.4, assume the load is equal to 5000 W at unity p.f. Compute the p.u. equivalent circuit. Assume Sb = 100 MVA, Vb = 345 kV.

4.6

A two-pole Y-connected generator rated at 13.8 kV, 20 MVA, 0.8 p.f. leading is running at 1800 rpm. The generator has a synchronous reactance of 8 Ω per phase (at 60 Hz) and a negligible armature resistance per phase. The generator is operated in parallel with an interconnected power network. Compute the following: (i) What is the torque angle of the generator at rated conditions? (ii) How many MW of power this generator can deliver?

4.7

A three-phase, eight-pole Y-connected synchronous generator rated at 220 MVA, 13.2 kV, 0.9 leading power factor is running at 1200 rpm. Its synchronous reactance is 0.8 Ω per phase at 60 Hz. The generator is fully loaded and supplies power to a network at rated voltage. Compute the generator voltage regulation.

4.8

The one-line diagram of a power grid is depicted in Figure 4.47. The data for the system is as follows: The transmission line between bus 2 and bus 3 is j10 Ω and between bus 3 and bus 5 is j6 Ω. G1 wind generating system: 25 MVA, 13.2 kV, and reactance of 0.20 per unit. G2 gas turbine generating system: 50 MVA, 20 kV, and reactance 0.20 per unit. Transformer T1: 100 MVA, 220 Y/13.8 Δ kV, and reactance of 10%. Transformer T2: 200 MVA, 220/20 kV, and reactance of 10%. Transformer T3: 100 MVA, 220 Y/22Y kV, and reactance of 10%. Compute the following: (i) Per unit impedance model for short-circuit studies. (ii) Per unit impedance model for power flow studies.

1 G1

T1

2

3

j10 Ω

j6 Ω

5

T3 4 Load

Figure 4.47

The system for Problem 4.8.

T2

6

G2

4.9

SMART POWER GRID SYSTEMS

The one-line diagram of a power grid is depicted in Figure 4.48. The data for the grid is given in the figure. Assume MVA base of 100 and base voltage of 13.2 kV; also assume the input reactance of the local power grid is 10% based on the transformer T1. The input reactance of all the sources is 7% with a base the same as their rating.

Infinite bus Local utility

HV bus

T1

CB

CB

CB

4.5 MW 2.2 MVAr

Utility EMS

1.2 MW 0.62 MVAr

CB

CB

13.2/3.3 kV 20 MVA Xt = 12%

7

1

To feeder

CB

T2

Net metering

8 0.05 + j0.5

242

10

CB

CB 0.04 + j0.4 9

11 0.9 MW 0.48 MVAr

CB

12

CB

CB

0.01 + j0.12

0.99 MW 0.54 MVAr

CB

CB CB

CB

13

CB

Gas turbine sync. gen. 1.8 MVA

3

CB 1.18 MW 0.62 MVAr

CB

CB

CB

CB

G

15

14

CB

0.04 + j0.45

CB

0.9 MW 0.48 MVAr

CB T3

CB

CB

0.03 + j0.32

CB

CB

0.96 MW 0.51 MVAr

0.04 + j0.5

T7

CB

CB

0.04 + j0.51

1.06 MW 0.56 MVAr

CB

CB CB

2

G Variable-speed wind turbine with DFIG (2 MW)

Micro power grid

DG EMS Local loads

16

17

T4 Local loads

T5

4

18 T6

Local loads

5

6

DC/AC

DC/AC

DC/AC

PV station

PV station

PV station

1 MW

1 MW

Figure 4.48 The system for Problem 4.9.

1 MW

PROBLEMS

243

TABLE 4.7 Transmission Line Data for Problem 4.9 Line

Resistance (Ω)

Series Reactance (Ω)

8–9 9–10 9–11 11–12 11–13 13–14 14–15

0.05 0.04 0.04 0.04 0.01 0.03 0.04

0.5 0.4 0.51 0.5 0.12 0.32 0.45

The system data are as follows: Transformer T1: 20 MVA, 33/13.2 kV, and 10% reactance. Transformer T2: 20 MVA, 13.2/3.3 kV, and 12% reactance. Transformer T3: 5 MVA, 3.3/460 V, and 6.5% reactance. Transformer T4, T5, and T6: 2 MVA, 3.3/460 V, and 6.5% reactance. Transformer T7: 5 MVA, 3.3/460 V, and 6% reactance. Transmission line impedance is given in Table 4.7. The local loads are 1 MVA at 0.85 power factor lagging. Perform the following: (i) Per unit model for power flow studies. (ii) Per unit model for short-circuit studies. 4.10

Consider microgrid grid given in Figure 4.49. Assume the following data: a. Transformers connected to the PV generating station are rated at 460 V Y grounded/13.2 kV Δ, with 10% reactance and 10 MVA capacity. The transformer connected to the power grid is 13.2/63 kV, 10 MVA, 10% reactance. Assume Sbase of 10 MVA and a voltage base of 460 V in the PV generator and the local power grid bus voltage has 5% tolerance. b. Assume the load on bus 4 is 1.5 MW at power factor of 0.9 lagging, on bus 5 is 2.5 MW at power factor of 0.9 lagging, on bus 6 is 1.0 MW at power factor of 0.95 lagging, on bus 7 is 2 MW at power factor of 0.95 leading, and on bus 8 is 1.0 MW and power factor of 0.9 lagging. c. The transmission line has a resistance of 0.0685 Ω/mile, with reactance of 0.40 Ω/mile and half of the line charging admittance (Y /2) of 11 × 10−6 Ω−1/mile. The line 4–7 is 5 miles, 5–6 is 3 miles, 5–7 is 2 miles, 6–7 is 2 miles, and 6–8 is 4 miles long. The transmission line model is given in Figure 4.50. d. Assume a PV generating station number 1 is rated at 0.75 MW and PV generating station number 2 is rated at 3 MW and PV generating stations are operating at unity power factors.

244

SMART POWER GRID SYSTEMS

63 kV

PV generating station 1

PV station DC/AC 460 V 2

1

13.2 kV

Local utility

P7 + jQ7

P5 + jQ5 13.2 kV

13.2 kV

7

5

Infinite bus

4 P4 + jQ4

13.2 kV

6

8 P6 + jQ6 3 460 V DC/AC PV station

P8 + jQ8

PV generating station 2

Figure 4.49 Photovoltaic (PV) microgrid of Problem 4.10.

R

j jXL

Y′/2

Y′/2

Per mile

Figure 4.50

Load

Y′/2 = jXC

A transmission line pie model.

Perform the following: (i) Per unit model for power flow studies. (ii) Per unit model for short-circuit studies. 4.11

Consider the PV power grid given in Figure 4.51. The PV1 has an internal resistance of 0.8 p.u. and inject 1 p.u. power into grid and PV2 has an internal resistance of 0.4 p.u. and inject 0.5 p.u. power into the grid. The load is 1 p.u. active and 1 p.u. reactive power. Perform the following: (i) Per unit model for short-circuit studies. (ii) Per unit model for power flow studies.

REFERENCES

245

PV DC/AC 1

3

j0.25 j0.2

All values in p.u. j0.4

2 DC/AC PV

Figure 4.51 A one-line diagram of Problem 4.11.

4.12

Compute the load factor for a feeder assuming that the maximum load is 8 MW and the average power is 6 MW.

4.13

Compute the load factor for a feeder for daily operation for one month assuming the same daily profile. Assume the average power is 170 MW and the peak is 240 MW.

4.14

If the feeder of Problem 4.13 is supplied from a wind source rated 80 MW and central power generating station rated at 500 MW, assume the capital cost of wind power is $500 per kW and the central station $100 per kW. Compute the EUC if the maintenance cost for the wind source is free, except for maintenance 1 ¢/kWh and central power generating station fuel and maintenance cost is 3.2 ¢/kWh. Give a figure for EUC from zero load factors to unity over 5 year’s utilization.

4.15

If the feeder of Problem 4.13 is supplied from 10 fuel cell sources rated for 2 MW and 20 microturbines rated at 6 MW. Assume the capital cost of the fuel cell is $1000 per kW and a microturbine is $200 per kW. Compute the energy cost (EUC) if the variable cost for the fuel cell is 15 ¢/kWh and the microturbines are 2.2 ¢/kWh, assuming 5 years of operation. Compute EUC from zero load factors to unity and plot it in a figure.

REFERENCES 1. Institute for Electric Energy. Homepage. Available at http://www.edisonfoundation. net/IEE. Accessed October 7, 2010. 2. MISO-MAPP Tariff Administration. Transition guide. Available at http://toinfo. oasis.mapp.org/oasisinfo/MMTA_Transition_Plan_V2_3.pdf. Accessed January 12, 2010. 3. Energy Information Administration. Official energy statistics from the US Government. Available at http://www.eia.doe.gov. Accessed October 29, 2010.

246

SMART POWER GRID SYSTEMS

4. Carbon Dioxide Information Analysis Center. Frequently asked global change questions. Available at http://www.cdiac.ornl.gov/pns/faq.html. Accessed September 29, 2009. 5. Shahidehpour, M. and Yamin, H. (2002) Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, Wiley/IEEE, New York/Piscataway, NJ. 6. Hirst, E. and Kirby, B. Technical and market issues for operating reserves. Available at http://www.ornl.gov/sci/btc/apps/Restructuring/Operating_Reserves.pdf. Accessed January 10, 2009. 7. Keyhani, A. and Miri, S.M. (1983) On-line weather-sensitive and industrial group bus load forecasting for microprocessor-based applications. IEEE Transactions on Power Apparatus and Systems, 102(12), 3868–3876. 8. Schweppe, F.C. (1978) Power systems 2000: hierarchical control strategies. IEEE Spectrum, 15(7), 42–47. 9. North American Electric Reliability Council. NERC 2008 long-term reliability assessment 2008–2017. Available at http://www.nerc.com/files/LTRA2008.pdf. Accessed October 8, 2010. 10. Wood, A.J. and Wollenberg, B.F. (1996) Power Generation, Operation, and Control, Wiley, New York. 11. Babcock & Wilcox Company (1975) Steam Its Generation & Use, 38th ed. Babcock & Wilcox, Charlotte, NC. 12. Nourai, A. and Schafer, C. (2009) Changing the electricity game. IEEE Power and Energy Magazine, 7(4), 42–47. 13. Ko, W.H. and Fung, C.D. (1982) VLSI and intelligent transducers. Sensors and Actuators, 2, 239–250. 14. De Almeida, A.T. and Vine, E.L. (1994) Advanced monitoring technologies for the evaluation of demand-side management programs. IEEE Transactions on Power Systems, 9(3), 1691–1697. 15. Adamiak, M. Phasor measurement overview. Available at http://phasors.pnl.gov/ Meetings/2004%20January/Phasor%20Measurement%20Overview.pdf. Accessed October 24, 2010. 16. Phadke, A.G. (1993) Computer applications in power. IEEE Power & Energy Society, 1(2), 10–15. 17. Schweppe, F.C. and Wildes, J. (1970) Power system static-state estimation, part I: exact model power apparatus and systems. IEEE Transactions on Volume, PAS-89(1), 120–125. 18. Ducey, R., Chapman, R., and Edwards, S. The U.S. Army Yuma Proving Ground 900-kVA photovoltaic power station. Available at http://photovoltaics.sandia.gov/ docs/PDF/YUMADOC.PDF. Accessed October 10, 2010. 19. Grainger, J. and Stevenson, W.D. (2008) Power Systems Analysis, McGraw Hill, New York. 20. Gross, A.C. (1986) Power System Analysis, Wiley, New York. 21. Majmudar, H. (1965) Electromechanical Energy Converters, Allyn and Bacon, Boston.

CHAPTER 5

SOLAR ENERGY SYSTEMS 5.1

INTRODUCTION

From the beginning of recorded history, humans have worshipped the sun. The first king of Egypt was “Ra, the sun god.” The sun god of justice for Mesopotamia was Shamash. In Hinduism, the sun god, Surya, is believed to be the progenitor of humanity. Apollo and Helios were the two sun divinities of Ancient Greece. The sun also figured prominently in the religious traditions of Zoroastrianism (Iran) and Buddhism (Asia), as well as in the Aztec (Mexico) and Inca (Peru) cultures.1 The sun’s energy is the primary source of energy for life on our planet. When the sun disappears from our universe, we will cease to exist.2 Solar energy is a readily available renewable energy; it reaches Earth in the form of electromagnetic waves (radiation). Many factors affect the amount of radiation received at a given location on Earth. These factors include location, season, humidity, temperature, air mass (AM), and the hour of the day. Insolation refers to exposure to the rays of the sun, that is, the word insolation has been used to denote the solar radiation energy received at a given location at a given time. The phrase incident solar radiation is also used; it expresses the average irradiance in watts per square meter (W/m2) or kilowatt per square meter (kW/m2). The surface of the Earth is coordinated with imaginary lines of latitude and longitude as shown in Figure 5.1a. Latitudes on the surface of the Earth are imaginary parallel lines measured in degrees. The lines subtend to the plane of Design of Smart Power Grid Renewable Energy Systems, Third Edition. Ali Keyhani. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/smartpowergrid3e

247

248

SOLAR ENERGY SYSTEMS

(a) 60 Latitude (degree)

40 20 0 –20 –40 –60 –180

–150

100

50

0

50

100

180

150

Longitude (degree) 640 – 900

900 – 1050

1500 – 1700

1050 – 1200

1200 – 1350

1900 – 2100

1700 – 1900

2100 – 2300

1350 – 1500 All values are in kWh/m2 > 2300

(b) 2400

2600

Global irradiance (kWh/m2)

2400

2200

2200 2000

2000 1800

1800

1600 1600

1400 1200

1400

1000 1200

800 0 20

1000 40 60

Longitude (degree)

80 150

100

50

0

–50

–100 –150

800

Latitude (degree)

Figure 5.1 (a) The global irradiation values for the world (kWh/m2).2,3 (b) The average northern hemisphere global irradiation source by longitude and latitude.4,5 (c) The average southern hemisphere global irradiation source by longitude and latitude. (d) The average world global irradiation source by regions.4,5 Region 1: Argentina, Chile; Region 2: Argentina, Chile; Region 3: Brazil, South Africa, Peru, Australia, Mozambique; Region 4: Indonesia, Brazil, Nigeria, Columbia, Kenya, Malaysia; Region 5: India, Pakistan, Bangladesh, Mexico, Egypt, Turkey, Iran, Algeria, Iraq, Saudi Arabia; Region 6: China, United States, Japan, Germany, France, United Kingdom, South Korea; Region 7: Russia, Canada, Sweden, Norway.

249

INTRODUCTION

(c) 2600

2400

Global irradiance (kWh/m2)

2400 2200

2200 2000

2000

1800 1800 1600 1600

1400 1200

1400

1000 1200

800 0 –20 –40 –60 –80

1000

0

–150 –100 –50

Longitude (degree)

50

100 150

800

Latitude (degree)

(d) 2800 Global irradiance (kWh/m2)

5 3

2400 2000

4

2

6

1600 1200

7 1

800 400

60° S

40° S

20° S 0° 20° N Latitude (degree)

Figure 5.1 (Continued)

40° N

60° N

250

SOLAR ENERGY SYSTEMS

the equator. The latitudes vary from 90 south (S) and 90 north (N). The longitudes are imaginary lines that vary from 180 east (E) to 180 west (W). The longitudes converge at the poles (90 N and 90 S). The radiation of the sun on the Earth varies with the location based on the latitudes. Approximately, the region between 30 S and 30 N has the highest irradiance as depicted in Figure 5.1. The latitude on which the sun directly shines overhead between these two latitudes depends on the time of the year. If the sun lies directly above the northern hemisphere, it is summer in the north and winter in the southern hemisphere. If it is above the southern hemisphere, it is summer in the southern hemisphere and winter in the north. Figure 5.1b is a plot showing the irradiance at different locations on the Earth marked by its latitude and longitude. The latitude varies from −90 to 90 , which is 90 N and 90 S, respectively. Similarly, the longitudes vary from −180 to 180 , which is 180 W and 180 E, respectively. The z-axis of the plot gives the irradiance in kWh/m2. The points on the z-axis convey the amount of irradiance. The colors on the plot represent the intensity of the irradiance as given in Figure 5.1b. The sun’s position as seen from Earth between latitudes 15 N and 35 N is the region with the most solar energy. This semiarid region, as shown in Figure 5.1b and c, is mostly located in Africa, the Western United States, the Middle East, and India. These locations have over 3000 hours of intense sunlight radiation per year. The region with the second largest amount of solar energy radiation lies between 15 N latitude and the equator and has approximately 2500 hours of solar energy per year. The belt between latitude 35 N and 45 N has limited solar energy. However, typical sunlight radiation is roughly about the same as the other two regions, although there are definite seasonal differences in both solar intensity and daily hours. As winter approaches, the solar radiation decreases; by midwinter, it is at its lowest level. The 45 N latitude and the region beyond experience approximately half of the solar radiation as diffused radiation. The energy of sunlight received by the Earth can be approximated to equal 10,000 times the world’s energy requirements.3 The sun’s radiation is in the form of ultraviolet, visible, and infrared energy as depicted in Figure 5.2. The majority of the energy is in the form of a shortwave used in the planet’s heat cycle, weather cycle, wind, and waves. A small fraction of the energy is utilized for photosynthesis in plants, and the rest of the solar energy emitted back into space. The solar energy reaching the atmosphere is constant—hence the term solar constant. The solar constant is computed to be in the range of 1.4 kW/m2, or 2.0 cal/cm2/min. Sunlight’s shorter wavelengths scatter over a wider area than the longer wavelengths of light. The scattering may be due to gas molecules, pollution, and haze. The blue and violet light has the maximum atmospheric scattering at sunrise and sunset without affecting the red rays of sunlight.

THE SOLAR ENERGY CONVERSION PROCESS: THERMAL POWER PLANTS

251

Electromagnetic spectrum Visible Cosmic rays Gamma rays 10–14

Microwave

X-rays Ultraviolet 10–10

TV Radio FM AM

Infrared 10–6

10–2

1

102

104

Wavelength in meters Energy

Figure 5.2 The electromagnetic spectrum.1,2 Source: Photo courtesy of the California Energy Commission.

5.2 THE SOLAR ENERGY CONVERSION PROCESS: THERMAL POWER PLANTS Augustin Mouchot constructed a parabolic mirror to channel the sun’s energy to power a steam engine in 1866.6 Todays’ thermal solar electric power plant also uses the sun’s heat energy to generate steam power for running the turbines. Concentrating solar power (CSP) systems have locating lenses designed for concentrating the sunlight into the receiver, which acts as a boiler to generate steam. The system uses a tracking control system for maximum efficiency. A number of concentrating technologies exists; a parabolic mirror that uses materials that are silver based or of polished aluminum is often used. Figure 5.3 depicts the CSP system located in the Mojave Desert in California, which uses a parabolic mirror. Solar power towers (see Figure 5.4) generate steam power by creating intense concentrated solar energy that is directed via a tower heat processing system.2 The system uses a large number of sun-tracking mirrors or parabolic reflectors. The number of mirrors depends on the system capacity. Another type of mirror used is called a heliostat (from Helios, the Greek word for sun). Earlier power towers used water/steam as the heat-transfer fluid. More recent advanced designs have used molten nitrate salt. A French physicist, Augustin-Jean Fresnel (1788–1827),8 developed the Fresnel lens, which has a large aperture and short focal length. Its construction requires less material than a conventional lens, and it allows for more light to pass through. The compact linear Fresnel reflector (CLFR) is used in solar power plants. The CLFR system uses a Fresnel lens and reflectors that are located on a single axis to concentrate the solar energy to generate steam. The CLFR uses thin mirror strips to focus high-intensity sunlight into a heat processing system. Flat mirrors are much cheaper to produce than parabolic ones, and they

252

SOLAR ENERGY SYSTEMS

Figure 5.3 The concentrated parabolic trough solar power system.7 Source: Photo courtesy of the California Energy Commission.

Electricity

Steam condenser Receiver Feedwater reheater

Generator Turbine Steam drum

Heliostats 1

Figure 5.4 A steam solar power generating station. Source: Photo courtesy of the California Energy Commission.

facilitate a greater number of reflectors for use in steam generation. Figure 5.5 depicts a CLFR power generating station. Another type of solar heat engine is the Stirling engine.1 It operates by cyclic compression with the expansion of the working fluid such as air or other gas. It uses two different temperature levels in its thermodynamic process converting heat energy into mechanical work. The Stirling solar dish engine system is an active area of research.4

PHOTOVOLTAIC MATERIALS

253

Figure 5.5 A compact linear Fresnel reflector (CLFR) concentrating solar power (CSP) technology.2

5.3

PHOTOVOLTAIC POWER CONVERSION

Solar cells convert the radiation energy directly to electric energy. Solar cells, also called photovoltaic (PV) cells, were developed by Carlson and Wronski in 1976.9 A PV module consists of a number of PV cells. When sunlight strikes the PV cell, electrons are freed from their atoms. The freed electrons are directed toward the front surface of the solar cell. This process creates a current flow that occurs between the negative and positive sides. The PV photon cell charge offers a voltage of 1.1–1.75 electron volt2 (eV2) with a high optical absorption. Figure 5.6 depicts a solar cell structure.2 Figure 5.6 represents a PV cell structure. A PV module connects a number of PV cells in series. You may think of a PV cell as a number of capacitors that are charged by photon energy of light. Figure 5.7 depicts how the irradiance energy of the sun is converted to electric energy using PV cells. 5.4

PHOTOVOLTAIC MATERIALS

The manufacture of PV cells is based on two different types of material: (1) a semiconductor material that absorbs light and converts it into electron–hole pairs and (2) a semiconductor material with junctions that separate photogenerated carriers into electrons and electron holes. The contacts on the front and back of the cells allow the current to the external circuit. Crystalline silicon cells (c-Si) are used for absorbing light energy in most semiconductors used in solar cells. Crystalline silicon cells are poor absorbers of light energy7; they have an efficiency in the range of 11–18% of that of solar cells. The most efficient monocrystalline c-Si cell uses laser-grooved, buried grid contacts, which allow for maximum light absorption and current collection.7 Each of the c-Si cells produces approximately 0.5 V. When 36 cells are connected in series, it creates an 18 V module. In the thin-film solar cell, the crystalline silicon wafer has a very high cost. Other common materials are amorphous silicon (a-Si) and cadmium telluride and gallium, which are another class of

254

SOLAR ENERGY SYSTEMS

Cell

Module

Array

Figure 5.6 A solar cell or photovoltaic cell structure.5

Sun

Front grid

Anti-reflection coating

Top contact

Base layer

Back contact

Figure 5.7 The structure of a photovoltaic cell.2

Space

PHOTOVOLTAIC CHARACTERISTICS

255

polycrystalline materials.7 The thin-film solar cell technology uses a-Si and a p-i-n single-sequence layer, where “p” is for positive and “n” for negative and “i” for the interface of a corresponding p- and n-type semiconductor.10 Thinfilm solar cells are constructed using lamination techniques, which promote their use under harsh weather conditions: they are environmentally robust modules. Due to the fundamental properties of c-Si devices, they may stay as the dominant PV technology for years to come. However, thin-film technologies are making rapid progress, and new material or process may replace the use of c-Si cells.9 Here we briefly introduce PV technology as it exists today. But as evolving technology, students and engineers should recognize that these advances will come from basic research in materials engineering and read the IEEE Spectrum to keep up with the developments in PV technology. Scientists at the National Renewable Energy Laboratory (NREL) are developing a new technique called hydride vapor phase epitaxy (HVPE), which holds the potential to produce cheaper, more efficient solar cells (https://www.nrel.gov/materialsscience/news.html). Students are also encouraged to obtain the up-to-date research information from the federal laboratory (https://www.nrel.gov/) dedicated to research, development, commercialization, and deployment of renewable energy and energy efficiency technologies. NREL conducts studies in all aspects of renewable energy and storage (https://www.nrel.gov/esif/labsenergy-storage.html) and learning about the capabilities, infrastructure, and research at the energy systems integration facility’s energy storage laboratory and practical issues with a solar resource for PV systems. We continue our discussion on how to develop models to study the integration of PV sources into the smart power grid system.

5.5

PHOTOVOLTAIC CHARACTERISTICS

As sun irradiance energy is captured by a PV module, the open-circuit voltage of the module increases.1,4 This point is shown in Figure 5.8 by Voc with the zero-input current. If the module is short-circuited, the maximum shortcircuit current can be measured. This point is shown in Figure 5.8 by Isc with zero-output voltage. The point on the I versus V characteristic where maximum power (PMPP) can be extracted lies at a current, IMPP, and the corresponding voltage point, VMPP. Typical data for PV modules are given in Table 5.1. This information is used to design PV strings and PV generating power sources. PV module selection criteria are based on a number of factors11: (1) the performance warranty, (2) module replacement ease, and (3) compliance with natural electrical and building codes. A typical silicon module has a power of 300 W with 2.43 m2 surface area; a typical thin film has a power of 69.3 W with an area of 0.72 m2. Hence, the land required by a silicon module is almost 35% less. Typical electrical data apply to standard test considerations (STC).

256

SOLAR ENERGY SYSTEMS

ISC

4

IMPP 3.5

Module current (A)

3 2.5 2 1.5 1 0.5 0 0

2

4

6

8

10

12

14

Module voltage (V)

16 18 VMPP

20

VOC

22

Figure 5.8 The operating characteristics of a photovoltaic module.5

TABLE 5.1 Voltage and Current Characteristics of Typical Photovoltaic Modules Module

Type 1

Type 2

Type 3

Type 4

Power (max), W Voltage at maximum power point (MPP), V Current at MPP, A VOC (open-circuit voltage), V ISC (short-circuit current), A Efficiency, % Cost, $ Width, in. Length, in. Thickness, in. Weight, lb

190 54.8 3.47 67.5 3.75 16.40 870.00 34.6 51.9 1.8 33.07

200 26.3 7.6 32.9 8.1 13.10 695.00 38.6 58.5 1.4 39

170 28.7 5.93 35.8 6.62 16.80 550.00 38.3 63.8 1.56 40.7

87 17.4 5.02 21.7 5.34 >16 397.00 25.7 39.6 2.3 18.3

For example, under STC, the irradiance is defined for a module with a typical value such as 1000 W/m2, spectrum AM 1.5, and a cell temperature of 25 C. Table 5.2 depicts the cell temperature characteristics of a typical PV module, and Table 5.3 depicts the maximum operating characteristics of a typical PV module.

257

PHOTOVOLTAIC CHARACTERISTICS

TABLE 5.2 Cell Temperature Characteristics of a Typical Photovoltaic Module Typical Cell Temperature Coefficient Tk (Pp) Tk (Voc) Tk (Isc)

Power Open-circuit voltage Short-circuit current

−0.47%/ C −0.38%/ C 0.1%/ C

TABLE 5.3 Maximum Operating Characteristics of a Typical Photovoltaic Module Limits Maximum system voltage Operating module temperature Equivalent wind resistance

600 V DC −40 to 90 C Wind speed: 120 mph

The PV fill factor (FF), as shown in Figure 5.9, is defined as a measure of how much solar energy is captured. This term is defined by PV module opencircuit voltage (Voc) and PV module short-circuit current (Isc): FF =

VMPP IMPP Voc Isc

(5.1)

4 VMPP, IMPP: Maximum power point

3.5

Module current (A)

3 2.5 Area B

Area A

2 Area B FF= Area A

1.5 1 0.5 0

0

2

4

6

8

10 12 14 Module voltage (V)

16

Figure 5.9 Photovoltaic module fill factor.

18

20

22

258

SOLAR ENERGY SYSTEMS

Current (A)

4 2 0 0 5

70 60

10 40

Voltage (V) 15

40 30

20

20 25

Temperature (°C)

10 30

0 –10

Figure 5.10 Three-dimensional I–V curve and temperature for a typical photovoltaic module.

and Pmax = FF Voc Isc = VMPP IMPP

(5.2)

In Figure 5.9, the maximum value for FF is unity. However, this value can never be attained. Some PV modules have a high FF. In the design of a PV system, a PV module with a high FF is used. For high-quality PV modules, FFs can be over 0.85. For typical commercial PV modules, the value lies around 0.60. Figure 5.10 depicts a three-dimensional display of a typical PV module and the fixed irradiance energy received by the module. As shown, a typical PV module characteristic is not only a function of irradiance energy, but it is also a function of temperature. 5.6

PHOTOVOLTAIC EFFICIENCY

The PV module efficiency, η, is defined as η=

VMPP IMPP PS

(5.3)

PHOTOVOLTAIC EFFICIENCY

259

where VMPPIMPP is the maximum power output, PMPP, and PS is the surface area of the module. The PV efficiency is defined as η = FF

Voc Isc ∞

(5.4)

P λ dλ

0

where P(λ) is the solar power density at wavelength λ. Figure 5.11 depicts a PV module consisting of 36 PV cells. If each cell is rated at 1.5 V, the module’s rated voltage is 54 V. Figure 5.12 depicts the basic configuration showing modules, strings, and an array. A string is designed by connecting a number of PV modules in series.

Rs ID1 Iph

ID2

IRp

Rp

I + V –

+

–

Figure 5.11 A photovoltaic module consisting of 36 photovoltaic cells.

String no.1 Module module

Array

String no.2

+

–

Figure 5.12 Basic configuration showing modules, strings, and an array.

260

SOLAR ENERGY SYSTEMS

A number of strings connected in parallel make an array. Two general designs of PV systems can be envisioned. Figure 5.13 depicts a PV design based on a central inverter. Figure 5.14 illustrates the utilization of multiple inverters. For a higher DC operating voltage, modules are connected in series. For a higher operating current, the modules are connected in parallel: n

Vj ; n number of series connected panels (5.5)

V series connected = j=1

For parallel connected panels, m

I parallel connected =

Ij ; m number of parallel connected panels j=1

(5.6)

. . . .

. . . .

Central inverter +

~ DC/AC ~

–

Figure 5.13 Central inverter for a large-scale photovoltaic power configuration.

DC Array 1

Inverter 1

Array 2

Inverter 2

Array 3

Inverter 3

Array 4

Inverter 4

Array 5

Inverter 5

Array 6

Inverter 6

Array 7

Inverter 7

Array 8

Inverter 8

Figure 5.14 General structure of photovoltaic arrays with inverters.

PHOTOVOLTAIC EFFICIENCY

261

Bypass diode

Blocking diode

Figure 5.15 Bypass and blocking diodes in a photovoltaic array.

In a PV system consisting of a number of arrays, all arrays must have equal exposure to sunlight: the design should place the modules of a PV system such that some of them will not be shaded. Otherwise, unequal voltages will result in some strings with unequal circulating current and internal heating producing power loss and lower efficiency. Bypass diodes are usually used between modules to avoid damage. Most new modules have bypass diodes in them as shown in Figure 5.15 to ensure longer life. However, it is complicated to replace built-in diodes if a diode fails in a panel. The PV industry, the American Society for Testing and Materials (ASTM),9 and the US Department of Energy have established a standard for terrestrial solar spectral irradiance distribution. An instrument called pyranometer5 measures the irradiance of a location. Figure 5.16 depicts irradiance in W/m2 per nanometer (nm) as a function of wavelength in nanometers (nm). The solar spectrum is the plot of the irradiance from the sun received at a particular location at a given temperature and AM flow. The intensity (W/m2/ nm) of the radiation is plotted as a function of wavelength (nm). The PV modules convert the radiation energy into electrical energy. The amount of energy produced by the PV module is directly proportional to the area of the module. The kW/m2/nm of the PV module is plotted as a function of wavelength and estimated temperature from the solar spectrum. The above curve is plotted with an AM of 1.5. The PV modules are tested at a nominal temperature (NT). The NT is used to estimate the cell temperature based on the ambient temperature as shown below: Tc = Ta +

NT−20 Kc

S

(5.7)

262

SOLAR ENERGY SYSTEMS

140

120 Power (kW/m2)

150 100

100 50

80

0 0

60 20 2500

40 2000

60

1500 80

Temperature (°C)

40

20

1000 500 100

0

Wavelength (nm)

Figure 5.16 Spectra for photovoltaic performance evaluation. Based on American Society for Testing and Materials and US Department of Energy National Renewable Energy Laboratory.12,13

The cell temperature, Tc, and the ambient temperature, Ta, are in degrees centigrade. The nominal operating temperature, NT, designates the cell temperature as tested by manufacturers; S is the solar insolation (kW/m2). Kc is a constant empirically computed from test data; it is in the range of 0.7–0.8. 5.7

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

The design of engineering systems is based on trial and error. Nevertheless, the design must demonstrate a clear understanding of the scientific and engineering principles behind the proposed design. In this sense, “trial and error” is for fine-tuning the final design. However, all engineering systems must be based on the underlying physical process. In concrete terms, if we are designing a PV generating system, we are using manufactured PV modules that have specific characteristics. However, in defining the overall objective of the PV systems we are putting into place, then some PV modules may satisfy those objectives, and some may not. Therefore, we need to test available PV modules against our design specifications. For a PV generating system, the first specification is the power requirement in kW or MW that we intend to produce. If the PV system is to operate as an independent power generating station, the rated load voltage is specified. For example, for a residential PV system, we may install from a few kilowatts of power to serve the residential loads at a nominal voltage of 120 and 207.8 V. If the residential PV is connected to the local utility, then the interconnection

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

263

voltage must be specified for the design of such a system. Other design restrictions may also apply to a design. Again, consider the case of designing a residential PV system. We already know that a PV module generates a DC voltage source and DC power. How high a DC voltage is safe in a residential system is determined by the electrical codes of a particular locality. In principle, we may want to design the residential PV system at a lower DC voltage and higher PV DC voltage for more involved power systems at commercial and industrial sites. Another design consideration for residential users may be the weight and surface area needed for a PV system. It is understood that the PV designer always seeks to design a PV system to satisfy the constraints of sites at the lowest installed and operating costs.14–17 Table 5.4 defines a few terms that we have discussed in this chapter. We will use these terms to introduce the design of PV systems. We know that all PV generating stations are designed based on connecting PV modules to generate the required power. (The terms PV module and PV panel are used interchangeably.) One PV module has a limited power rating;

TABLE 5.4 Photovoltaic Design Terms Terms

Abbreviations

Descriptions

String voltage

SV

Power of a module String power Number of strings Number of arrays Surface area of a module Total surface area Array power

PM SP NS NA SM TS AP

Number of modules Total number of modules

NM TNM

Array voltage for maximum power point tracking Array current for maximum power point tracking Array maximum power point Number of converters Number of rectifiers Number of inverters

VAMPP

String voltage for series-connected modules Power produced by a module Power generated in one string Number of strings per array Number of arrays in a design Surface area of a module Total surface area Array power is generated by connecting a number of strings in parallel Number of modules per string Total number of modules in all arrays put together The operating voltage for maximum power point tracking of an array The operating current for maximum power point tracking of an array The maximum operating power of an array Total number of DC/DC converters Total number of AC/DC rectifiers Total number of inverters

IAMPP PAMPP NC NR NI

264

SOLAR ENERGY SYSTEMS

therefore, to design a higher power rating, we construct a string by connecting a number of PV modules in series: SV = NM × Voc

(5.8)

where SV defines the string voltage and Voc is the open-circuit voltage of a module. As an example, if the number of modules is five and the open-circuit voltage of the module is 50 V, we will have SV = 5 × 50 = 250 V DC The above voltage may be a high voltage in a residential PV system. We can think of a PV cell, a module, or an array as a charged capacitor. The amount of charge of a PV system is a function of solar irradiance. At full sun, the highest amount of charge is stored that will generate the highest open-circuit voltage for a PV system. In general, the open-circuit voltage of a PV panel for a residential system might be set at a voltage lower than 250 V DC. Local electric codes govern the rated open-circuit voltage. In general, the open-circuit string voltage is less than 600 V DC for a commercial and industrial system at this time. However, higher DC voltage designs of PV systems are considered for higher power PV sites. The string power, SP, is the power that can be produced by one string: SP = NM × PM

(5.9)

where NM is the number of modules and PM is the power produced by a module. For example, if a design uses four PV modules, each rated 50 W, then the total power produced by the string is given as SP = 4 × 50 = 200 W As we discussed, for producing higher rated power from a PV generating station, we can increase the string voltage. Also, we can connect some strings in parallel and create an array. Therefore, the array power, AP, is equal the number of string times the string power: AP = NS × SP

(5.10)

If the number of strings is 10 and each string is producing 200 W, we will have AP = 10 × 200 = 2000 W

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

265

The DC power produced by an array is the function of the sun’s position and irradiance energy received by the array. An array can be located on roofs or free-standing structures. The array power is processed by a converter to extract maximum power from the sun’s irradiance energy. Because the transfer of DC power over a cable at low voltages results in high power losses, the array power is either converted to AC power by using a DC/AC inverter. In some applications, the array power is converted to a higher DC voltage power to produce higher AC voltage and to process higher rated power. For the maximum power out of an array, the maximum power point tracking (MPPT) method is used. The MPPT method locates the point on the trajectory of power produced by an array where the array voltage and array current are at its maximum point and the maximum power output for the array. The array maximum power point (MPP) is defined as PMPP = VAMPP × IAMPP

(5.11)

where VAMPP is the array voltage at MPPT and IAMPP is the array current at its MPPT. An array is connected to either an inverter or a boost converter, and the control system operates the array at its MPPT. The final design of a PV system is based on the MPP operation of a PV array generating station. Keeping in mind, however, the converter control system is designed to locate the maximum operating point based on generated array voltage and array current, thus accommodating the changing irradiance energy received by an array as the sun’s position changes during the day. The inverter output voltage is controlled by controlling the inverter amplitude modulation index. To process the maximum power by an inverter, the amplitude modulation index, Ma, should be set at a maximum value without producing the unwanted harmonic distortion. The value of Ma is set less than one and in the range of 0.95 to produce the highest AC output voltage. Example 5.1 Design a photovoltaic (PV) system to process 10 kW of power at 230 V, 60 Hz single-phase AC. Determine the following: (i) Number of modules in a string and number of strings in an array. (ii) Inverter specification and one-line diagram. The PV module data is given in Table 5.5. Solution Table 5.5 depicts the voltage and current characteristics of a typical PV module. The load voltage is specified as 230 V single-phase AC. To acquire

266

SOLAR ENERGY SYSTEMS

TABLE 5.5 The Voltage and Current Characteristics of a Typical Photovoltaic Module Power (max) Maximum voltage, PMPP

300 W

Voltage at maximum power point (MPP), VMPP Current at MPP, IMPP Voc (open-circuit voltage) Isc (short-circuit current)

50.6 V 5.9 A 63.2 V 6.5 A

maximum power from the PV array, we select a modulation index of Ma = 0.9. The inverter input voltage is given by Vidc =

2Vac Ma

Vidc =

2 × 230 = 361 4 V 09

(5.12)

The inverter is designed to operate at the maximum power point tracking of PV array. Therefore, the number of modules to be connected in series in a string is given by NM =

Vidc VMPP

(5.13)

where VMPP is the voltage at the maximum power point of PV of the module: NM =

361 4 ≈7 50 6

The string voltage is given as SV = NM × VMPP

(5.14)

Using this module, string voltage (see Table 5.5) for this design is SV = 7 × 50 6 = 354 2 V (i) The power generated by one string is given by SP = NM × PMPP where PMPP is the nominal power generated at the maximum power point tracking.

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

267

The power generated by a string for this design is given as kW per string = 7 × 300 = 2100 W To calculate the number of strings for a 10 kW PV system, we divide the PV power rating by power per string NS =

AP SP

(5.15)

where NS is the number of string and AP is the array power and SP is string power. For this design we have NS =

10 × 103 =5 2100

Therefore, we have five strings and one array to generate 10 kW of power. Table 5.6 presents the PV specifications for a 10 kW generating station. (ii) In the final design, the inverter should be rated such that it can process generation of 10 kW and supply the load at 230 V AC from its array at its maximum power point tracking. Based on the PV module of Table 5.5, the string voltage is specified as Vidc = 354 2 V and the modulation index is given as follows: Ma =

2Vac Vidc

Ma =

2 × 230 = 0 92 354 2

Let us select a switching frequency of 6 kHz. Therefore, the frequency modulation index is provided by Mf =

fS 6000 = 100 = 60 fe

TABLE 5.6 Photovoltaic Specifications for a 10 kW Generating Station Modules per String 7

Strings per Array

Number of Arrays

String Voltage (V)

5

1

354.2

268

SOLAR ENERGY SYSTEMS

TABLE 5.7 Inverter Specifications Input Voltage, Vidc (V) 354.2

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

10

230

0.92

100

1

2 DC bus

Amplitude modulation index = 0.92

AC bus

DC/AC

10 kW load

Inverter 354.2 V

230 V

5

PV array: five string, seven modules per string

Figure 5.17 The one-line diagram in Example 5.1.

Table 5.7 presents the inverter specifications. The one-line diagram is given in Figure 5.17. Example 5.2 Design a photovoltaic (PV) system to process 500 kW of power at 460 V, 60 Hz three-phase AC using PV data of Example 5.1. Determine the following: (i) (ii) (iii) (iv)

Number of modules in a string and number of strings in an array. Inverter and boost specification. The output voltage as a function and total harmonic distortion. The one-line diagram of this system.

Solution The load is 500 kW rated at 460 V AC. Based on the voltage of the load and an amplitude modulation index of 0.9, we have the following input DC voltage for a three-phase inverter:

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

Vidc = =

269

2 2VLL 3Ma 2 2 × 460 3×0 9

(5.16) = 835 V

We will limit the maximum string voltage to 600 V DC. Therefore, we can use a boost converter to boost the string voltage to 835 V. If we select the approximate string voltage of 550 V, we will have the following: (i) The number of modules in a string is given by Vstring 550 ≈11 = VMPP 50 6 where VMPP is the voltage of a module at maximum power point tracking. The string power, SP, can be computed as SP = NM × PMPP Using a module rated at 300 W, we will have SP = 11 × 300 = 3300 W The string voltage is given as SV = NM × VMPP Therefore, the string voltage, SV, for this design is SV = 11 × 50 6 = 556 6 V If we design each array to generate a power of 20 kW, then the number of strings, NS, in an array is given by NS =

Power of one array Power of one string

NS =

20 =6 33

The number of arrays, NA, for total power generation is NA =

PV generation Power of one array

(5.17)

270

SOLAR ENERGY SYSTEMS

TABLE 5.8 Photovoltaic Specifications Modules per String 11

Strings per Array

Number of Arrays

String Voltage (V)

6

25

556.6

Therefore, NA =

500 kW = 25 20 kW

Table 5.8 gives the PV specifications. (ii) The inverters should be rated to withstand the output voltage of the boost converter and should be able to supply the required power. The inverter is rated at 100 kW with an input voltage of 835 V DC and the amplitude modulation index of 0.9. The output voltage of the inverter is 460 V AC. The number of inverters, NI, needed to process a generation of 500 kW is given by NI =

PV generation Power of one inverter

(5.18)

Therefore, NI =

500 =5 100

Hence, we need to connect five inverters in parallel to supply the load of 500 kW if a switching frequency is set at 5.04 kHz. Therefore, the frequency modulation index, Mf, is given by Mf =

fS 5040 = 84 = 60 fe

Table 5.9 presents the inverter specifications. The number of boost converters needed is the same as the number of arrays, which is 25, and the power rating of each boost converter is 20 kW. The boost converter input voltage is equal to the string voltage: Vi = 556 6 V TABLE 5.9 Inverter Specifications Number of Inverters 5

Input Voltage Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

835

100

460

0.90

84

271

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

The output voltage of the boost converter is equal to the inverter input voltage: Vidc = Vo = 835 V The duty ratio of the boost converter is given by D = 1−

Vi Vo

D = 1−

556 6 = 0 33 835

Table 5.10 presets the boost converter specifications for a generation of 500 kW. (iii) With a frequency modulation of 84, the harmonic content of the output voltage was computed from a simulation testbed using a fast Fourier method; it is tabulated in Table 5.11. The output line-to-neutral voltage as a function of time is Vac =

460 2 3

sin 2π60 t +

0 01 460 2 sin 2π × 3 × 60 t × 100 3

+

0 02 460 2 0 460 2 sin 2π × 5 × 60 t + sin 2π × 7 × 60 t × × 100 100 3 3

+

0 03 460 2 sin 2π × 9 × 60 t × 100 3

= 376sin 2π60 t + 0 037sin 6π60 t + 0 075sin 10π60 t + 0 113sin 18π60 t

TABLE 5.10 Boost Converter Specifications for a Generation of 500 kW Number of Boost Converters 25

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

556.6

20

835

0.33

TABLE 5.11 Harmonic Content of Line-to-Neutral Voltage Relative to the Fundamental Third Harmonic 0.01%

Fifth Harmonic

Seventh Harmonic

Ninth Harmonic

0.02%

0

0.03%

272

SOLAR ENERGY SYSTEMS

DC bus 1

Duty ratio = 0.33

DC bus

AC bus Amplitude modulation index = 0.9

DC/DC Boost

Three-phase inverter

DC/DC Boost

5

25

556.6 V 25 PV arrays: 6 string 11 modules per string

Figure 5.18

1

DC/AC

2

DC/DC

DC/AC

Boost 20 kW each

Three-phase inverter 100 kW each

835 V

500 kW

460 V

The one-line diagram of Example 5.2.

The total harmonic distortion is given by THD =

harmonic 2 =

0 012 + 0 022 + 02 + 0 032 = 0 04

(iv) The one-line diagram is given in Figure 5.18. Students should note that the above example is an analysis problem for demonstrating the basic concept. For proper operation of Example 5.2, the PV generating station must be connected to a local grid for the proper frequency of operation or have a battery storage system for DC/AC converter to operate as an uninterruptible power supply (UPS) system.

Example 5.3 Design a photovoltaic (PV) system to process 1000 kW of power at 460 V, 60 Hz three-phase AC using the PV data given in Table 5.12. Determine the following: (i) Number of modules in a string, number of strings in an array, number of arrays, surface area for PV, the weight of PV, and cost. (ii) DC/AC inverter and boost converter specifications and the one-line diagram of the system.

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

273

TABLE 5.12 Photovoltaic Data for Example 5.3 Module

Type 1

Power (max), W The voltage at MPP, V Current at MPP, A Voc (open-circuit voltage), V Isc (short-circuit current), A Efficiency, % Cost, $ Width, in. Length, in. Thickness, in. Weight, lb

190 54.8 3.47 67.5 3.75 16.40 870.00 34.6 51.9 1.8 33.07

Solution The load is 1000 kW rated at 460 V AC. Based on the voltage of the load and an amplitude modulation index of 0.85, the input DC voltage for a threephase inverter is Vidc =

2 2VLL 3Ma

=

2 2 × 460 3 × 0 85

= 884 V

We will limit the maximum voltage that a string is allowed to have to 600 V. Therefore, we use a boost converter to boost the string voltage to 884 V. (i) If we select approximate string voltage of 550 V, the number of modules in a string, NM, is given as NM =

Vstring 550 = ≈ 10 VMPP 54 8

where VMPP is the voltage at MPP of the PV module. SP = NM × PMPP where PMPP is the power generated by a PV module at MPP. SP = 10 × 190 = 1900 W And the string voltage, SV, is SV = NM × VMPP

274

SOLAR ENERGY SYSTEMS

Therefore, the string voltage, SV, for this design is SV = 10 × 54 8 = 548 V If each array is to have a rating of 20 kW, the number of strings, NS, in an array is NS =

AP SP

NS =

20 = 11 19

The number of arrays, NA, for this design is NA =

PV generation Power of one array

NA =

1000 = 50 20

The total number of PV modules, TNM, is given by the product of the number of modules per string, the number of strings per array, and the number of arrays: TMS = NA∗NS∗NM

(5.19)

TNM = 10 × 11 × 50 = 5500 The surface area of one module, SM, is given by the product of its length and width: SM =

34 6 × 51 9 = 12 5 ft2 144

The total surface area, TS, is therefore given by the total number of modules and the surface area of each module: TS = 5500 × 12 5 = 68,750 ft2 =

68,750 = 1 57 acre 43,560

The total cost of PV modules is given by the product of the number of PV modules and the cost of one module: The total cost = 5500 × 870 = $4 78 million

275

THE DESIGN OF PHOTOVOLTAIC SYSTEMS

The total weight of PV modules is given by the product of the number of PV modules and the weight of one module: The total weight = 5500 × 33 07 = 181 ,885 lb Table 5.13 presents the PV specifications for 1000 kW generating station. (ii) The inverters should be rated to withstand the output voltage of the boost converter and should be able to supply the required power (see Table 5.14). Selecting an inverter rated at 250 kW, we will have the number of inverters, NI, needed to process the generation of 1000 kW as given by NI =

PV generation Power of one inverter

NI =

1000 =4 250

Hence, we need to connect four inverters in parallel to supply the load of 1000 kW. For a switching frequency of 5.40 kHz, the frequency modulation index is given by Mf =

fS 5400 = 90 = 60 fe

TABLE 5.13 Photovoltaic Specifications for 1000 kW Generating Station Modules per String 10

Strings per Array

Number of Arrays

String Voltage (V)

Total Area (ft2)

Total Weight (lb)

Total Cost (million $)

11

50

548

68,750

181,885

4.78

TABLE 5.14 Inverter Specifications Number of Inverters 4

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

884

250

460

0.85

90

276

SOLAR ENERGY SYSTEMS

The number of boost converters needed is the same as the number of arrays, which is 50. Selecting a boost converter rating of 20 kW and the boost converter input voltage to be equal to the string voltage, Vi = 548 V The output voltage of the boost converter is equal to the inverter input voltage: Vidc = Vo = 884 V The duty ratio of the boost converter is given by Vi Vo 548 = 0 38 D = 1− 884

D = 1−

Table 5.15 presents the boost converter specifications. The one-line diagram is given in Figure 5.19. TABLE 5.15 Boost Converter Specifications Number of Boost Converters

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

548

20

884

0.38

50 DC bus 1

Duty ratio = 0.38 DC/DC Boost

2

DC bus 1

Three-phase inverter

DC/DC Boost

DC/DC 548 V 50 PV arrays: 11 string array, 10 modules per string

Figure 5.19

Boost 20 kW each

1

DC/AC

2

50

AC bus Amplitude modulation index = 0.85

4

50

1000 kW

DC/AC 884 V

Three-phase inverter 250 kW each

The one-line diagram of Example 5.3.

460 V

THE MODELING OF A PHOTOVOLTAIC MODULE

5.8

277

THE MODELING OF A PHOTOVOLTAIC MODULE

As we discussed, a commercial PV module is constructed from a number of PV cells. A PV cell is constructed from a p-n homojunction material. A p-type doped semiconductor is joined with an n-type doped semiconductor, and a p-n junction is formed. If the p-type and the n-type semiconductors have the same bandgap energy, a homojunction is formed. The homojunction is a semiconductor interface, a phenomenon that takes place between layers of similar semiconductor material. These types of semiconductors have equal bandgaps, and they normally have different doping. The absorption of photons of energy generates DC power. Figure 5.20 shows a PV cell. When irradiance energy of the sun is received by the module, it is charged with electric energy. The model of a PV cell is similar to that of a diode and can be expressed by the well-known Shockley–Read equation. The PV module can be modeled by a single exponential model. Figure 5.21 presents a PV circuit model. Equation (5.20) presents the single exponential model with the current source and the output voltage as a function of the temperature. The current source, Iph, in parallel with the shunt resistance, Rsh, describes the PV model. The current flowing through the shunt resistance is designated as IRsh. The output DC voltage, V, is in series with the internal resistance, Rs. The PV model of Figure 5.21 also depicts the power loss through current, ID1, circulating through the diode: I = Iph − I0 exp

q V + IRs nc AkT

−1 −

V + IRs Rsh

(5.20)

Irradiance

PV module PV model Iph I0 A Rs Rsh Temperature

Figure 5.20 The modeling of a photovoltaic module.

V

278

SOLAR ENERGY SYSTEMS

Rs

I +

Iph

+ –

ID1

IRsh

V

Rsh

–

Figure 5.21 The single exponential model of a photovoltaic module.

Other parameters are the diode quality factor (A); the number of cells in the module, nc; Boltzmann’s constant (k), 1.38 × 10−23 J/K; the electronic charge (q), 1.6 × 10−19 C; and the ambient temperature (T) in Kelvin. Equation (5.20) is nonlinear, and its parameters Iph, Io, Rs, Rsh, and A are functions of temperature, irradiance, and manufacturing tolerance. We can use numerical methods and curve fitting to estimate the parameters from test data provided by manufacturers. The estimation of PV array model is quite involved and is presented in Reference18. 5.9

THE MEASUREMENT OF PHOTOVOLTAIC PERFORMANCE

A PV module at a maximum constant level of irradiance can produce 1000 W/m2, which is also termed as one sun. The power output of the PV module is calibrated based on its exposure to the sun. Table 5.16 depicts the irradiance in W/m2. The sun irradiance energy is calibrated for a PV array system based on the angle of incidence. This data is used to operate the PV array at its MPP. In the next section, we will discuss how the power converters use digital controllers to operate a PV generating station at its MPP. 5.10 THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY First, let us review the maximum power transfer in a resistive circuit. Consider the circuit of Figure 5.22. Assume a voltage source with an input resistance, Rin. This source is connected to a load resistance, RL. TABLE 5.16 Sun Performance Versus Incident Irradiance Sun Performance One Sun 0.8 Sun 0.6 Sun 0.4 Sun 0.2 Sun

Incident Irradiance 1000 W/m2 800 W/m2 600 W/m2 400 W/m2 200 W/m2

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

279

Rin I + V

–

RL

Figure 5.22 A DC source with a resistive load.

The current supplied to the load is I=

V Rin + RL

(5.21)

The power delivered to the load RL is P = I 2 RL P = V2

(5.22) RL

Rin + RL

(5.23)

2

Differentiating with respect to RL,

dP = V2 dRL

Rin + RL

dP Rin −RL = V2 dRL Rin + RL

2 dRL

−RL

dRL Rin + RL 3

d Rin + RL dRL

2

4

(5.24)

Setting the above to zero, we can calculate the operating point for the maximum power. The MPP can be delivered to the load when RL = Rin. A PV module output power is the function of irradiance solar energy. Figure 5.23 depicts the output power in W/m2 at various irradiances as a function of module current and output voltage. The PV system should be operated to extract the maximum power from its PV array as the environmental conditions change about the position of the sun, cloud cover, and daily temperature variations. The equivalent circuit model of a PV array depicted in Figure 5.21 can be presented during its power transfer mode to a load RL as shown in Figure 5.24. Figure 5.24 presents the circuit model for a PV source by a current source that has a shunt resistance, Rsh, and series resistance, Rs.19 The shunt resistance has a large value and series resistance is very small. RL represents the

280

SOLAR ENERGY SYSTEMS

(a) 1

50 °C

0.8 75 °C PV module current (p.u.)

1.4

25 °C

100 mW/cm2

0.7

1.2 PMPP 1

0.6 0.8

0.5 0.4

0.6

0.3

0.4

0.2 0.2

0.1 0 0

PV module output power (p.u.)

0.9

0.1

0.2

0.3 0.4 0.5 0.6 PV module voltage (p.u.)

0.7

0.8

0 0.9

(b) 1 25 °C

100 mW/cm2

PV module current (p.u.)

0.8 0.7

1.4 1.2

PMPP 75 mW/cm2

1

0.6 0.8

0.5 0.4

50 mW/cm2

0.6

0.3

0.4

0.2 0.2

0.1 0 0

PV module output power (p.u.)

0.9

0.1

0.2

0.3 0.4 0.5 0.6 PV module voltage (p.u.)

0.7

0.8

0 0.9

Figure 5.23 (a) The PV output current versus output voltage and output power as a function of temperature variation. (b) The output power in W/m2 at various irradiances as a function of module current and output voltage.10

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

281

Rs

IPh

Rsh

RL

Figure 5.24 A photovoltaic model and its load. Req

VPV

+ –

RL

Figure 5.25 A simple voltage source equivalent circuit model of a photovoltaic array.

load resistance. In Figure 5.24, RL is the reflected load because in practice the load is connected to the converter side if the PV operates as a stand-alone. When the PV is connected to the power grid, the load is based on the injected power to the power grid. Figure 5.25 depicts the equivalent model of a PV with a voltage source model replacing a current source model. Figure 5.23 shows that the characteristics of a PV module are highly nonlinear. The input impedance of a PV array is affected by irradiance variation and temperature. The corresponding output power is also shown in Figure 5.23. Figure 5.26 depicts the PV energy processing system using a boost converter to step up the voltage and an inverter to convert the DC power to AC. To achieve maximum power transfer from the PV array, the input impedance of the PV generator must match the load. The MPPT control algorithm seeks to operate the boost converter at a point on the PV array current and voltage characteristics where the maximum output power can be obtained. For a PV power generating station, the control algorithm computes the dP/ dV > 0 and dP/dV < 0 to identify if the peak power has been obtained. Figure 5.27 depicts the control algorithm. If the PV system is to supply power to DC loads, then a DC/AC inverter is not needed. Depending on the application, a number of designs of a PV system can be proposed. When the PV system is to charge a battery storage system, the PV system can be designed as depicted in Figure 5.28 using a boost converter or as in Figure 5.29 using a buck converter. In this design again, the MPPT is accomplished using an MPPT control algorithm depicted in Figure 5.27.

282

SOLAR ENERGY SYSTEMS

DC/DC boost converter

RL RL

PWM

IPV

VPV

RL

DC/AC inverter

Digital MPPT controller

PWM

Idc Vdc

Iload

Digital controller Vload

Figure 5.26 A photovoltaic energy processing using a boost converter to step up the voltage and an inverter.

Measured power PPV

IPV PPV = IPV VPV VPV

Pmax = IMPP VMPP

IPV

PWM signal

Increase IPV current

Converter

VPV

Figure 5.27 A maximum power point tracking control algorithm. PV arrays Boost converter L

D SW C

Vo

Battery

PWM switching for generating duty cycle VPV IPV

Digital controller Pmax = IMPPVMPP MPPT current control

IPV VPV

Figure 5.28 Maximum power point tracking using only a boost converter.

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

283

PV arrays Buck converter

L SW

Cin

D

C

Vo

Battery

PWM switching for generating duty cycle

VPV IPV

Digital controller MPPT and current control

Pmax = IMPPVMPP

IPV VPV

Figure 5.29

Maximum power point tracking using a buck converter.

Figure 5.30 depicts the design of a PV generating system and MPPT using an inverter when the PV generating station connected to a local utility. Again, the digital controller tracks the PV station output voltage and current and computes the MPPT point according to the control algorithm of Figure 5.27. The control algorithm issues the pulse width modulation (PWM) switching policy to control inverter current such that the PV station operates at its MPP. However, the resulting control algorithm may not result in a minimum total harmonic distortion. The design presented in Figure 5.27 has two control loops. The first control loop is designed to control the DC/DC converter, and the second control loop can control the total harmonic distortion and output voltage. The tracking of MPPT may not be optimum when the MPPT control performed as part of the inverter as shown in Figure 5.30. In this type of MPPT, the current to the inverter flows through all modules in the string. However, the I–V curves may not be the same, and some strings will not operate at their MPPT. Therefore, the resulting energy capture may not be as high, and some energy will be lost in such systems. Figure 5.31 depicts a PV generating station with a battery storage system when the PV system is connected to the local utility. The DC/DC converter and its MPPT are referred to a charge controller. The charge controllers have a number of functions. Some charge controllers are used to detect the variations in the current–voltage characteristics of a PV array. MPPT controllers are necessary for a PV system to operate at a voltage close to MPP to draw maximum available power as shown in Figure 5.27. The charge controllers

284

SOLAR ENERGY SYSTEMS PV arrays DC/AC Inverter SW1+ SW1–

D1+

SW2+

D2+

D1–

SW2–

D2–

a

n

D3+

SW3+ SW3–

D3– c

b

PWM switching

c b a

IPV

Local power grid

V Digital controller PWM current control and MPPT IPV

Pmax = IMPPVMPP

I

VPV

VPV

Figure 5.30 A photovoltaic generating station operating at maximum power point tracking when the photovoltaic system connected to a local power grid.

also perform battery power management. For normal operation, the controllers control the battery voltage, which varies between the acceptable maximum and minimum values. When the battery voltage reaches a critical value, the charge controller function is to charge the battery and protect the battery from an overcharge. This control accomplished by two different voltage thresholds, namely, battery voltage and PV module voltage. At lower voltage, typically 11.5 V, a controller switches the load off and charges the battery storage system. At higher voltage, usually 12.5 V for 12 V battery storage system charge, a controller switches the load to the battery. The control algorithm adjusts the two voltage thresholds depending on the battery storage system. DC/DC MPPT PV charge controllers facilitate standardization of integration of PV system for use in a local storage system. The system of Figure 5.31 can also be used as a stand-alone microgrid that can deliver high-quality power for a UPS. Example 5.4 Design a microgrid of photovoltaic (PV) system rated at 50 kW of power at 220 V, 60 Hz single-phase AC using a boost converter and singlephase DC/AC inverter. Use the data given in Tables 5.17–5.19 for your design. Determine the following: (i) Number of modules in a string for each PV type, number of strings in an array for each PV type, number of arrays and surface area, weight, and cost for each PV type. (ii) Boost converter and inverter specifications and the one-line diagram of this system.

Power (W)

MPPT control PV voltage and current

DSP controller

Irradiance

Voltage (V)

Gating signal

AC voltage and current

Gating signal

+ VL R I

D

I

R

+

+

L

Vin

–

iL

+ –

D

–

+

V

i

Temperature

DSP controller

S

C

CB

+

R VO

–

–

Infinite bus

–

+

CB

CB

Local power grid

+ –

–

DC/AC inverter

DC/DC boost converter

Transformer model Local load

Nonlinear photovoltaic model R

RS

RL

V C(ψC)

IC

RT

ψC

+ + –

CS

CL IC

VC –

Non-linear battery model

Figure 5.31 A photovoltaic generating station operating at maximum power point tracking with a battery storage system when the photovoltaic system connected to a local power grid.

285

286

SOLAR ENERGY SYSTEMS

TABLE 5.17 Typical Photovoltaic Modules Module

Type 1

Type 2

Type 3

Type 4

Power (max), W Voltage at maximum power point (MPP), V Current at MPP, A Voc (open-circuit voltage), V Isc (short-circuit current), A Efficiency, % Cost, $ Width, in. Length, in. Thickness, in. Weight, lb

190 54.8 3.47 67.5 3.75 16.40 870.00 34.6 51.9 1.8 33.07

200 26.3 7.6 32.9 8.1 13.10 695.00 38.6 58.5 1.4 39

170 28.7 5.93 35.8 6.62 16.80 550.00 38.3 63.8 1.56 40.7

87 17.4 5.02 21.7 5.34 >16 397.00 25.7 39.6 2.3 18.4

TABLE 5.18 Single-Phase Inverter Data Inverter

Type 1

Type 2

Type 3

Type 4

Power Input voltage DC Output voltage AC Efficiency

500 W 500 V

5 kW 500 V max

15 kW 500 V

4.7 kW 500 V

230 VAC/60 Hz at 2.17 A

230 VAC/60 Hz at 27 A

220 VAC/60 Hz at 68 A

230 VAC/60 Hz at 17.4 A

Min. 78% at full load 15.5 5 5.3 9 lb

97.60%

>94%

96%

315 mm 540 mm 191 mm 23 lb

625 mm 340 mm 720 mm 170 kg

550 mm 300 mm 130 mm 21 kg

Length Width Height Weight

TABLE 5.19 Typical Boost Converters Input Voltage (V) 24–46 24–61 24–78 24–78 24–98 80–158 80–198 80–298 200–600

Output Voltage (V)

Power (kW)

26–48 26–63 26–80 26–80 26–100 82–160 82–200 82–300 700–1000

9.2 12.2 11.23 13.1 12.5 15.2 14.2 9.5 20.0

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

287

Solution The load is 50 kW rated at 220 V AC. Based on the voltage of the load and an amplitude modulation index of 0.9, the input DC voltage for the inverter is Vidc =

2Vac = Ma

2 × 220 = 345 V 09

(i) If we select string voltage, SV, of 250 V, the number of modules is NM =

String voltage VMPP

where VMPP is the voltage at MPP of the PV module: NM =

250 ≈ 5 for type 1 54 8

=

250 ≈ 10 for type 2 26 3

=

250 ≈ 9 for type 3 28 7

=

250 ≈ 15 for type 4 17 4

The string voltage, SV, is given as SV = NM × VMPP Therefore, the string voltage, SV, for this design is SV = 5 × 54 8 = 274 V for type 1 = 10 × 26 3 = 263 V for type 2 = 9 × 28 7 = 258 3 V for type 3 = 15 × 17 4 = 261 V for type 4 Selecting the 15.2 kW boost converter from Table 5.19, the number of boost converters, NC, is NC =

PV generation Boost converter power rating

NC =

50 ≈4 15 2

288

SOLAR ENERGY SYSTEMS

Therefore, the design should have four arrays: each with its boost converter. The array power, AP, is AP =

PV generation Number of arrays

AP =

50 = 12 5 kW 4

The string power, SP, is given as SP = NM × PMPP where PMPP is the power generated by the PV module at MPP: SP = 5 × 190 = 0 95 kW for type 1 = 10 × 200 = 2 0 kW for type 2 = 9 × 170 = 1 53 kW for type 3 = 15 × 87 = 1 305 kW for type 4 The number of strings, NS, is given by NS =

Power per array Power per string

NS =

12 5 = 14 for type 1 0 95

=

12 5 = 7 for type 2 2

=

12 5 = 9 for type 3 1 53

=

12 5 = 10 for type 4 1 305

The total number of modules, TNM, is given by TNM = NM × NS × NA TNM = 5 × 14 × 4 = 280 for type 1 = 10 × 7 × 4 = 280 for type 2 = 9 × 9 × 4 = 324 for type 3 = 15 × 10 × 4 = 600 for type 4

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

289

The surface area, TS, needed by each PV type is given by the product of the total number of modules and the length and the width of one PV module:

TS =

280 × 34 6 × 51 9 = 3492 ft2 for type 1 144

=

280 × 38 6 × 58 5 = 4391 ft2 for type 2 144

=

324 × 38 3 × 63 8 = 5498 ft2 for type 3 144

=

600 × 25 7 × 39 6 = 4241 ft2 for type 4 144

The total weight needed for each PV type is the product of the number of modules and the weight of one module: The total weight = 280 × 33 07 = 9260 lb for type 1 = 280 × 39 00 = 10, 920 lb for type 2 = 324 × 40 70 = 13, 187 lb for type 3 = 600 × 18 40 = 11, 040 lb for type 4 The total cost for each PV type is the product of the number of modules and the cost of one module: The total cost = 280 × 870 = $243, 600 for type 1 = 280 × 695 = $194, 600 for type 2 = 324 × 550 = $178, 200 for type 3 = 600 × 397 = $238, 200 for type 4 Table 5.20 presents the PV specifications for each PV type. (ii) The boost converter rating is

The boost converter power rating = The boost converter rating =

PV generation Number of converters 50 = 12 5 kW 4

290

SOLAR ENERGY SYSTEMS

TABLE 5.20 The Photovoltaic Specifications for Each Photovoltaic Type Total Total Number Number String Total Area Weight Cost Number of of the of the of the Voltage of Modules per of Strings PV PV (lb) PV ($) PV (ft2) (V) per Array Arrays String Type 1 2 3 4

5 10 9 15

14 7 9 10

4 4 4 4

274 263 258.3 261

3,492 4,391 5,498 4,241

9,260 10,920 13,187 11,040

243,600 194,600 178,200 238,200

If the boost converter output voltage selected is equal to Vidc = Vo = 345 V, then the input voltage is equal to string voltage: Vi = 274 V for type 1 Vi = 263 V for type 2 Vi = 258 3 V for type 3 Vi = 261 V for type 4 The duty ratio of the boost converter is given by

D = 1−

Vi Vo

D = 1−

274 = 0 205 for type 1 PV 345

= 1−

263 = 0 237 for type 2 PV 345

= 1−

258 3 = 0 251 for type 3 PV 345

= 1−

261 = 0 243 for type 4 PV 345

Table 5.21 presents the boost converter specifications. The inverters should be rated to withstand the output voltage of the boost converter and should be able to supply the required power. Let us design with each inverter having a rating of 10 kW. The input voltage of the inverter is Vidc = 345 V with an amplitude modulation index of 0.90. The output voltage of the inverter is at 220 V AC.

291

THE MAXIMUM POWER POINT OF A PHOTOVOLTAIC ARRAY

TABLE 5.21 Boost Converter Specifications PV Type 1 2 3 4

Number of Boost Converters

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

4 4 4 4

274 263 258.3 261

12.5 12.5 12.5 12.5

345 345 345 345

0.205 0.237 0.251 0.243

The number of inverters, NI, to process a generation of 50 kW is given by NI =

PV generation Power of one inverter

NI =

50 =5 10

Hence, we need to connect five inverters in parallel to supply the load of 50 kW. Of course, we can also use one inverter with a higher rating to convert the DC power to AC. Naturally, many other designs are also possible. Selecting a switching frequency of 5.1 kHz, the frequency modulation index will be given as Mf =

fS 5100 = 85 = 60 fe

Students can compute the total harmonic distortion. Table 5.22 presents the inverter specifications. Figure 5.32 presents a one-line diagram of the PV generating station. The selection of the type of PV system may be based on the weight and cost of the system. For residential and commercial systems with existing roof structures, the PV modules that have minimum weight are normally selected. The selection of the boost converter is based on the power rating of the boost converter and its output voltage.

TABLE 5.22 Inverter Specifications Number of Inverters 5

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

345

10

220

0.90

85

292

SOLAR ENERGY SYSTEMS

DC bus 1

DC bus DC/DC Boost

2

1

2

3 5 DC/AC 4

4 DC/DC

Figure 5.32

1

One-phase inverter

DC/DC Boost

PV array

AC bus

DC/AC

DC/DC Boost 3

Amplitude modulation index = 0.90

Boost 12.5 kW each

345 V

One-phase inverter 10 kW each

50 kW

220 V

The one-line diagram of Example 5.4.

The boost converter must be rated at the minimum output voltage of the PV system and the required DC input voltage of the inverter. The amplitude modulation index is selected to be less than one, but close to one for processing the maximum power of the DC source to the AC power. The frequency modulation index selected is based on the highest sampling time recommended by the manufacturer of the inverter to limit the total harmonic distortion. The number of strings and the number of modules in the string are based on the rating of the input voltage of the boost converter. The number of boost converters and inverters is based on the required output power of the PV generating station.

5.11 A BATTERY STORAGE SYSTEM The capacity of a battery is rated in ampere-hours (Ah). The Ah measures the capacity of a battery to hold energy. One ampere-hour means that a battery can deliver one amp for one hour. Based on the same concept, a 110 Ah battery can produce 10 A for 11 hours. However, after a battery is discharged for an hour, the battery will need to be charged for longer than one hour. It is estimated it will take 1.25 Ah to restore the battery to the same state of charge. Battery performance also varies with temperature, battery type, and age. Lead–acid battery technology is well established and is a widely adopted energy source for various power industries. Recent advances in the design of the deep-cycle lead–acid battery have promoted the use of battery storage systems when rapid discharge and charging are required. For example, if the load requires a 900 Ah bank, a number

A BATTERY STORAGE SYSTEM

293

of battery storage systems can be designed. As a first design, three parallel strings of deep-cycle batteries rated at 300 Ah can be implemented. The second design can be based on two strings of deep-cycle 450 Ah batteries. Finally, the design can be based on a single large industrial battery. Lead–acid batteries are designed to have approximately 2.14 V per cell. For an off-the-shelf 12 V battery, the voltage rating is about 12.6–12.8 V. The fundamental problem is that if a single cell in a string fails, the entire storage bank will rapidly discharge beyond the required discharge level; this will permanently destroy the bank. It is industry practice not to install more than three parallel battery strings. Each string is monitored to ensure equal charging and discharging rates. If a storage bank loses its equalization, it will result in accelerated failure of any weak cells and the entire storage bank. The battery characteristics change with age and charging and discharging rates. It is industry practice not to enlarge an old battery bank by adding new battery strings. As the battery ages, the aging is not uniform for all cells. Some cells will establish a current flow to the surrounding cells, and this current will be difficult to detect. If one cell fails, changing the resistance in one battery string, the life of the entire string can be reduced substantially. Therefore, the storage system will fail. By paralleling several strings, the chances of unequal voltages across the strings increase; thus, for optimized battery storage systems, the system should be designed using a single series of cells that are sized for their loads. The battery capacity can be estimated for a given time duration by multiplying the rated load power consumption in watts by the number of hours that the load is scheduled to operate. This results in energy consumption in watthours (or kilowatt-hours [kWh]), stated as kWh = kV Ah

(5.25)

For example, a 60 W light bulb operating for one hour uses 60 Wh. However, if the same light bulb is supplied by a 12 V battery, the light it will consume is 5 Ah. Therefore, to compute Ah storage required for a given load, the average daily usage in W should be divided by the battery voltage. As another example, if a load consumes 5 kWh per day from a 48 V battery storage system, we can determine the required Ah by dividing the watt-hours by the battery voltage. For this example, we will need 105 Ah. However, because we do not want to discharge the battery more than 50%, the battery storage needed should be 210 Ah. If this load has to operate for 4 days, the required capacity is 840 Ah. If the battery cabling is not properly insulated from earth, the capacitive coupling from the DC system with earth can cause stray current flow from the DC system to underground metallic facilities, which will corrode the underground metallic structure. Table 5.23 presents typical battery storage systems. Battery energy storage is still expensive for large-scale stationary power applications under the current electric energy rate. However, the battery storage system is an essential

294

SOLAR ENERGY SYSTEMS

TABLE 5.23 Typical Battery Storage Systems Class Class Class Class Class Class Class Class

1 2 3 4 5 6 7 8

34–40 Ah 70–85 Ah 85–105 Ah 95–125 Ah 180–215 Ah 225–255 Ah 180–225 Ah 340–415 Ah

12 V 12 V 12 V 12 V 12 V 12 V 6V 6V

technology for the efficient utilization of an intermittent renewable energy system such as wind or PV in the integration of renewable energy in electric power systems. Utility companies are interested in the large-scale integration of a battery storage system in their substations as community storage to capture the high penetration of solar energy and wind in their distribution system. The community storage system with the ramping capability of at least an hour can be utilized in power system control. The technology of the storage system is an essential consideration for utility companies because the installed energy storage system can be used as a spinning reserve. The storage system must be effectively and efficiently scheduled for the charging, discharging, and rest time of each string. Therefore, the battery storage system requires extensive monitoring and control.20,21 5.12 A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY The rapid electrification of the automotive industry has a significant impact on storage technology and its use in stationary power. At present, nickel metal hydride (NiMH) batteries are used in most electric and hybrid electric vehicles available to the public. Lithium-ion batteries have the best performance of the available batteries. For the cost of a battery energy storage system in a PV power station, students are encouraged to read the up-to-date reports from the National Renewable Energy Laboratory at https://www.nrel.gov/grid/ assets/. The large battery storage system is constructed from single-cell batteries13,20,21 and is considered a multicell storage system. The performance of multicell storage is a function of output voltage, internal resistance, cell connections, the current discharge rate, and cell aging.13,20,21 Single-cell battery technology is rapidly making new advances, for example, a new lithium-ion battery is being developed. The price of a single-cell battery has also been dropping, in comparison with the regular lead–acid battery where 12 cells are internally connected in a 12 V battery. The single-cell batteries can be individually connected and reconfigured. Also, because the individual single-cell rate of charging and discharging can be monitored, the health

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

295

and performance of all cells in a string can be evaluated. Figure 5.33 depicts three single cells in a string. Figure 5.34 depicts a battery storage system consisting of two strings of three single cells that are connected in parallel making an array. Table 5.24 presents the energy densities and power density of two singlecell batteries. Table 5.25 presents the energy density and the cost of a storage system of lead–acid batteries as compared with single-cell batteries. In all batteries, the performance will change with repeated charging at the current discharge rates. As expected, the higher the current discharge rate, the lower

–

Cell no.1

Cell no.2

+

Cell no.3

Load

Figure 5.33 Three single cells in a string.

Cell no.1

Cell no.2

Cell no.3

Cell no.4

Cell no.5

Cell no.6

–

+

Load

Figure 5.34 Two strings of three single cells connected in parallel. TABLE 5.24 Comparison of Battery Energy Density and Power Density13 Application/ Battery Type NiMH hydride Lithium ion

Energy Density (Wh/kg)

Energy Stored (kWh)

The Fraction of Usable Energy (%)

65 130

40–50 40–50

80 80

296

SOLAR ENERGY SYSTEMS

TABLE 5.25 The Energy Density and the Cost of a Storage System Type Standard Thin film NiMH hydride Lithium ion

Wh/kg

Wh

Weight kg (1)

$/kg

$/kWh

$/kW

25 20 45 65

1875 1000 1800 1170

75 50 40 18

2.5 4.0 22.5 45

100 200 500 700

9.35 10.0 45.0 41.0

the remaining capacity and output voltage, and the higher the internal resistance. However, the reduced capacity as the result of a higher discharge current rate will be recovered after the battery system is allowed to rest before the next discharge cycle. Therefore, the design and optimization of a multicell storage system require an understanding of the storage system’s discharge performance under various operating conditions. Furthermore, if the battery storage system is to be used as a community storage system in a power grid distribution system, dynamic models of the battery systems are needed. The dynamic model of a storage system will facilitate dispatching power from intermittent green energy sources. Example 5.5 Design a microgrid with the load of 1000 kW rated at 460 V AC and connected to the local power grid at 13.2 kV using a transformer rated at 2 MVA, 460 V/13.2 kV, and 10% reactance. For the emergency loads, microgrid needs 200 kWh storage systems to be used for 8 hours a day. Data for a three-phase inverter is given in Table 5.26. The data for the photovoltaic (PV) system is presented in Example 5.4.

TABLE 5.26 Three-Phase Inverter Data Inverter

Type 1

Type 2

Type 3

Type 4

Power Input voltage DC Output voltage AC Efficiency

100 kW 900 V

250 kW 900 V max

500 kW 900 V

1 MW 900 V

Depth Width Height Weight

660 VAC/60 Hz 660 VAC/60 Hz 480 VAC/60 Hz 480 VAC/60 Hz Peak efficiency 96.7% 30.84 57 80 2,350 lb

Peak efficiency 97.0% 38.2 115.1 89.2 2,350 lb

Peak efficiency 97.6% 43.1 138.8 92.6 5,900 lb

Peak efficiency 96.0% 71.3 138.6 92.5 12,000 lb

297

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

Determine the following: (i) The ratings of PV arrays, converters, inverters, storage systems, and a single-line diagram of this design based on the minimum surface area. Also, compute the cost, weight, and square feet area of each PV type and give the results in a table. (ii) Per unit (p.u.) model for the design.

Solution (i) The load is 1000 kW rated at 460 V AC. Based on the voltage of the load and an amplitude modulation index of 0.9, we have the following input DC voltage for the inverter: Vidc =

2 2VLL 3Ma

=

2 2 × 460 3×0 9

= 835 V

For an inverter rated at 250 kW, the total number of inverters, NI, for the processing of 1000 kW is given as NI =

PV generation Rating of inverters

NI =

1000 =4 250

For this design, four inverters should be connected in parallel. If we select a switching frequency of 5.04 kHz, the frequency modulation index is Mf =

fS 5040 = 84 = 60 fe

Table 5.27 presents the inverter specifications. Students should recognize that other designs are also possible. The input DC voltage of PV specifies the output AC voltage of inverters. Table 5.28 gives the data for each PV type. TABLE 5.27 Inverter Specifications Number of Inverters 4

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

835

250

460

0.90

84

298

SOLAR ENERGY SYSTEMS

TABLE 5.28 A Photovoltaic System Design Based on Photovoltaic Type PV Type 1 2 3 4

Surface Area of One Module (ft2)

Power Rating (W)

Area per Unit Power (ft2/W)

12.47 15.68 16.97 7.07

190 200 170 87

0.066 0.078 0.100 0.081

As noted in Table 5.28, the PV module of type 1 requires the minimum of surface area. Selecting PV type 1 and string open-circuit voltage of 550 V DC, the number of modules, NM, is NM =

String voltage VMPP

where VMPP is the voltage at the maximum power point of the PV module from the PV data: NM =

550 ≈ 10 for type 1 PV 54 8

The string voltage, SV, under load is given as SV = NM × VMPP SV = 10 × 54 8 = 548 V The string power, SP, is given as SP = NM × PMPP where PMPP is the power generated by the PV module at the maximum power point. SP = 10 × 190 = 1 9 kW for type 1 If we design each array to generate a power of 20 kW, then the number of strings, NS, is given by NS =

Power of one array Power of one string

NS =

20 = 11 19

299

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

The number of arrays, NA, is given by NA =

PV generation Power of one array

NA =

1000 = 50 20

The total number of PV modules, TNM, in an array is given by TNM = NM × NS × NA where NM is a number of modules in a string, NS is the number of strings, and NA is the number of arrays in a PV station: NA = 10 × 11 × 50 = 5500 for PV module of type 1 The total surface area needed, TS, for type 1 PV module is TS =

5500 × 34 6 × 51 9 = 68, 586 ft2 = 1 57 acre 144

The total weight, TW, needed for a type 1 PV module is the product of the number of modules and the weight of each module: TW = 5500 × 33 07 = 181,885 lb The total cost for a PV module is the product of the number of modules and the cost of each module: Total cost = 5500 × 870 = $ 4 78 million for PV module type 1 Table 5.29 presents the PV specifications. The output voltage of the boost converter, Vo, is the same as the input voltage of the inverter, Vidc: Vo = Vidc = 835 V

TABLE 5.29 The Photovoltaic Specifications Number of Modules PV Type per String 1

10

Number of Strings per Array 11

Total Total Area Weight Total Cost Number String of the of the PV Voltage of the of PV (ft2) PV (lb) (million $) (V) Arrays 50

548

68,586

181,885

4.78

300

SOLAR ENERGY SYSTEMS

The boost input voltage, Vi, is the same as the string voltage: SV = Vi = 548 V The duty ratio of the boost converter is given by D = 1−

Vi Vo

For this design, it is D = 1−

548 = 0 34 835

We need one boost converter for each array. Therefore, the number of boost converters is 50, and each is rated 20 kW. Table 5.30 presents the boost converter specifications. Table 5.31 presents a number of batteries for storing 200 kWh of energy. In storage design, we need to limit the number of batteries in a string and limit the number of arrays to three. These limitations are imposed on lead–acidtype batteries to extend the life of the storage system. We select the class 6 batteries that are rated at 255 Ah at 12 V. In this design, three batteries per string and three strings in each array are used. The string voltage, SV, of the storage system is SV = 3 × 12 = 36 V The string energy stored, SES, in each battery is given by the product of the Ah and the battery voltage: SES = 255 × 12 = 3 06 kWh TABLE 5.30 Boost Converter Specifications Number of Boost Converters

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

548

20

835

0.34

50

TABLE 5.31 Battery Array Specifications Battery Class

Number of Batteries per String

Number of Strings per Array

Number of Arrays

String Voltage (V)

Energy Stored per Array (kWh)

6

3

3

8

36

27.54

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

301

Each array has nine batteries. Therefore, the array energy stored, AES, is given as AES = 9 × 3 06 = 27 54 kWh The number of arrays, NA, needed to store 200 kWh is given by NA =

Total energy Energy in each array

NA =

200 ≈8 27 54

Because we have eight storage arrays, we use one buck–boost converter for each array storage system. We need a total of eight buck–boost converters. The buck–boost converters are used to charge/discharge the battery storage system. In this design, the buck–boost converter input is 835 V of the DC bus, and its output must be 36 V DC to charge the battery storage system. If the storage systems are to be used for 8 hours, they can be discharged to 50% of their capacity. Therefore, they can be used to supply 100 kWh. The power, P, supplied by the storage system is given by P=

kWh hour

P=

100 = 12 5 kW 8

The array power, AP, rating is given by AP =

Power Number of arrays

AP =

12 5 = 1 56 kW 8

Let us select a buck–boost converter rated at 1.56 kW. The duty ratio is given by D=

Vo Vi + Vo

D=

36 = 0 04 835 + 36

Table 5.32 presents the buck–boost converter specifications. Figure 5.35 depicts the one-line diagram of the PV system.

302

SOLAR ENERGY SYSTEMS

TABLE 5.32 Buck–Boost Converter Specifications Number of Buck– Boost Converters

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

835

1.56

36

0.04

8

1

DC bus 1

1 AC bus DC/AC 1

DC/DC

DC/ DC Buck–boost

Boost 50 PV arrays: 11 strings array, 10 modules per string

Three-phase inverter

460 V/13.2 kV 2 MVA X = 10%

8 battery arrays: 3 strings array, 3 batteries per string

50 50 DC/DC 548 V

Boost 20 kW each

8 DC/ DC Buck–boost 1.56 kW each

Local power grid

13.2 kV 1000 kW

36 V 4 DC/AC Three-phase inverter 250 kW each

835 V

460 V

Figure 5.35 The one-line diagram of Example 5.5.

(ii) To compute the per unit system, we let the base volt-ampere be Sb = 1 MVA. The base values for the system of Figure 5.35 are as follows: The base value of watt-hours is, therefore, Eb = 1 MWh. The base voltage on the utility side is 13.2 kV. The base voltage on the low-voltage side of the transformer is 460 V. The new p.u. reactance of the transformer on the new 1 MVA base is given by

Xp u

trans new

= Xp u

Xp u

trans new

=0 1×

trans old

×

Sb new Vb old × Sb old Vb new

1 13 2 × 2 13 2

2

2

= 0 05 p u

The per unit power, Pp.u., a rating of the inverters is given by Pp u = Pp u -inverter =

Power rating Sb 250 × 103 1 × 106

= 0 25 p u

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

303

And the per unit power rating of the boost converters is Pp u

boost

=

20 × 103 1 × 106

= 0 020 p u

The base voltage of the DC side of the inverter is 835 V. Therefore, the p.u. voltage of the DC side, Vp.u., of the inverter is Vp u =

835 = 1p u 835

Because the base voltage of the low-voltage side of the boost converter is the same as the rated voltage, the per unit value is one p.u. The same is true for the storage system. Similarly, the per unit power for the buck–boost and the energy storage system can be computed: Pp u

buck–boost

=

Energy stored =

1 56 × 103 1 × 106

= 0 001 p u

27 54 × 103 1 × 106

= 0 027 p u

Figure 5.36 presents the per unit model of the system. Example 5.6 Design a microgrid of photovoltaic (PV) system using the data in Table 5.17 assuming the total load is 500 kW at 460 V, 60 Hz threephase AC.

DC bus 1

1 –

+

DC/DC

1 DC/ DC

Boost 50 PV arrays: 11 strings per array, 10 modules per string

+

+–

Three-phase inverter

Buck–boost

8 battery arrays: 3 strings per array, 3 batteries per string 0.027 p.u. of energy per array 8

50 –

1 AC bus DC/AC

50 DC/DC

DC/ +– DC Buck–boost Vb = 36 V 0.001 p.u. each

Boost Vb = 548 V 0.02 p.u. Vb = 835 V each

Sb = 1 MVA

X = 0.05 p.u.

Local power grid

1 p.u. 1 p.u. Vb = 13.2 kV 4 DC/AC Three-phase inverter 0.25 p.u. each

Vb = 460 V

Figure 5.36 The per unit model of the system outlined in Example 5.5.

304

SOLAR ENERGY SYSTEMS

For each type of PV system, determine the following: (i) The number of modules in a string, number of strings in an array, number of arrays, weight, and surface area. (ii) The boost converter and inverter specifications and the one-line diagram.

Solution The load is 500 kW rated at 460 V AC. Based on the voltage of the load and an amplitude modulation index of 0.9, the input DC voltage for the threephase inverter from Equation (5.16) is 2 2 × 460 3×0 9

= 835 V

(i) Limiting the maximum voltage for a string to 600 V DC, a boost converter has to be used to boost the string voltage. If we select string voltage, SV, to 550 V, the number of modules, NM, from Equation (5.13) is

NM =

550 ≈ 10 for type 1 54 8

=

550 ≈ 21 for type 2 26 3

=

550 ≈ 20 for type 3 28 7

=

550 ≈ 32 for type 4 17 4

The string voltage, SV, from Equation (5.14) is SV = 10 × 54 8 = 548 V for type 1 = 21 × 26 3 = 552 V for type 2 = 20 × 28 7 = 574 V for type 3 = 32 × 17 4 = 557 V for type 4

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

305

The string power from Equation (5.9) is SP = 10 × 190 = 1 9 kW for type 1 = 21 × 200 = 4 2 kW for type 2 = 20 × 170 = 3 4 kW for type 3 = 32 × 87 = 2 784 kW for type 4 If we design each array to generate power of 20 kW, then the number of strings, NS, from Equation (5.15) is 20 = 11 for type 1 19 20 = = 5 for type 2 42 20 = 6 for type 3 = 34 20 = 8 for type 4 = 2 784

NS =

The number of arrays, NA, is given by Equation (5.17): NA =

500 = 25 20

The total number of PV modules, TNM, is given by Equation (5.19): TNM = 10 × 11 × 25 = 2750 for type 1 = 21 × 5 × 25 = 2625 for type 2 = 20 × 6 × 25 = 3000 for type 3 = 32 × 8 × 25 = 6400 for type 4 The total surface area, TS, needed by each PV type can be computed from the number of modules and each module’s area in square feet: 2750 × 34 6 × 51 9 = 34,294 ft2 = 0 787 acre for 144 2625 × 38 6 × 58 5 = 41,164 ft2 = 0 944 acre for = 144 3000 × 38 3 × 63 8 = 50,907 ft2 = 1 169 acre for = 144 6400 × 25 7 × 39 6 = 45,232 ft2 = 1 038 acre for = 144

TS =

type 1 type 2 type 3 type 4

306

SOLAR ENERGY SYSTEMS

The total weight is Total weight = 2750 × 33 07 = 90,943 lb for type 1 = 2625 × 39 00 = 102,375 lb for type 2 = 3000 × 40 70 = 122,100 lb for type 3 = 6400 × 18 40 = 117,760 lb for type 4 The total cost for each design is Total cost = 2750 × 870 = $2 39 million for type 1 = 2625 × 695 = $1 82 million for type 2 = 3000 × 550 = $1 65 million for type 3 = 6400 × 397 = $2 54 million for type 4 Table 5.33 presents the PV specifications for each PV type. (ii) We need to use one boost converter for each array. The total number of converters is 25—each can be selected with 20 kW rating. The nominal output voltage of the boost converter is the same as the input of the inverter: Vidc = Vo = 835 V The input voltage, Vi, of the boost converter is equal to the string voltage: Vi = 548 V for type 1 Vi = 552 V for type 2 Vi = 574 V for type 3 Vi = 557 V for type 4 TABLE 5.33 The Photovoltaic Specifications for Each Photovoltaic Type Total Total Area Weight Number of Number of Number String of the Voltage of the of Strings Modules PV PV (ft2) PV (lb) (V) Type per String per Array Arrays 1 2 3 4

10 21 20 32

11 5 6 8

25 25 25 25

548 552 574 557

34,294 41,164 50,907 45,232

90,943 102,375 122,100 117,760

Total Cost of the PV (million $) 2.39 1.82 1.65 2.54

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

307

The duty ratio of the boost converter is given by D = 1−

Vi Vo

D = 1−

548 = 0 34 for type 1 PV 835

= 1−

552 = 0 34 for type 2 PV 835

= 1−

574 = 0 31 for type 3 PV 835

= 1−

557 = 0 33 for type 2 PV 835

Table 5.34 presents the boost converter specifications. The inverters should be rated to withstand the output voltage of the boost converter and should be able to supply the required power. We can select each inverter having a rating of 100 kW, the input voltage of the inverter to be Vidc = 835 V with amplitude modulation index of 0.9, and the output voltage of the inverter at 460 V AC. The number of inverters, NI, from Equation (5.18), needed to process a generation of 500 kW is given by NI =

500 =5 100

Hence, we need to connect five inverters in parallel to supply the load of 500 kW. If a switching frequency of 10 kHz is selected to limit the total harmonic distortion, then the frequency modulation index is given by Mf =

fS 10,000 = 166 67 = 60 fe

TABLE 5.34 Boost Converter Specifications PV Type 1 2 3 4

Number of Boost Converters

Input Voltage, Vi (V)

Power Rating (kW)

Output Voltage, Vo (V)

Duty Ratio, D

25 25 25 25

548 552 574 557

20 20 20 20

835 835 835 835

0.34 0.34 0.31 0.33

308

SOLAR ENERGY SYSTEMS

TABLE 5.35 Inverter Specifications Number of Inverters 5

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

835

100

460

0.90

166.67

DC bus

DC bus 1 DC/DC Boost 2

AC bus Amplitude modulation index = 0.90

1

1

DC/AC

2 DC/DC

Three-phase inverter

Boost

5 Boost

DC/AC 25

25

Three-phase inverter 100 kW each

DC/DC PV array

Boost 20 kW each

835 V

500 kW

460 V

Figure 5.37 The one-line diagram of Example 5.6.

Table 5.35 presents the inverter specifications. The one-line diagram of the system is shown in Figure 5.37. Example 5.7 Assume a residential house total load is 7.5 kW from 11 pm to 8 am and 15 kW for the remaining 24 hours. Determine the following: (i) Plot load cycle for 24 hours. (ii) Total kWh energy consumption for 24 hours. (iii) If the solar irradiance is 0.5 sun for the 8 hours daily operation, what is the roof space needed to generate adequate kWh for 24 hours’ operation? (iv) Assume the maximum kWh to be used during the night is 40% of the total daily load. Search the Internet to select a battery storage system and give your design data.

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

309

Solution (i) The load is 7.5 kW for 9 hours (from 11:00 pm to 8:00 am) and 15 kW for 15 hours (from 8:00 am to 11:00 pm). The load cycle is as given in Figure 5.38. (ii) The total kWh energy consumption for 24 hours is the area under the curve of the daily load cycle and is given by kWh = kW × hours. Therefore, the energy consumption = 7.5 × 9 + 15 × 15 = 292.5 kWh. (iii) The type 1 PV is selected because it needs the minimum area per unit power produced. The amount of power produced by type 1 PV (see Table 5.17) is equal to 190 W per module for 1 sun. Therefore, the energy produced for 0.5 suns for 8 hours is given by 0.5 × 190 × 8 = 0.76 kWh. The number of modules, NM, needed is given by NM =

Total energy demand Energy of one panel

NM =

292 5 ≈ 385 0 76

The surface area, SM, of one module is given by width × length: SM = 34 6 × 51 9 144 = 12 47 ft2 The total area, TS, for 292.5 kWh is given by the product of the number of modules and the area of one module: TS = 385 × 12 47 = 4801 11 ft2 (iv) The energy used during the night is 40% of the total energy. Therefore, the energy demand for one night = 0.4 × 292.5 = 117 kWh.

Power (kW)

15 10 5 0

3

6

9

12

15

18

21

24

Time (hours)

Figure 5.38 Plot of the daily load cycle for Example 5.7.

310

SOLAR ENERGY SYSTEMS

From Table 5.23, a class 6 battery storage system is chosen to store the kWh needed for the night. For battery storage conservation, the batteries should not be discharged more than 50% of their capacity. The energy stored per battery is given by amp-hours × voltage (Ah × V). Therefore, the energy stored in one battery =255 × 12 = 3.06 kWh. The number of batteries, NB, needed is given by 2 × Energy demand Energy stored per battery 2 × 117 = 77 NB = 3 06

NB =

We can use three batteries in a string; the maximum number of strings in an array is equal to three. Therefore, the maximum number of batteries in the array is 3 × 3 = 9. The number of arrays of the battery is given by Total number of batteries Number of batteries per array 77 =8 NA = 9

NA =

Example 5.8 Consider a microgrid with 2 MW. The system operates as part of a microgrid connected to the local utility at 13.2 kV. The local utility uses the following design data: (a) The transformer specifications are 13.2 kV/460 V, 2 MVA, and 10% reactance and 460 V/220 V, 20 kVA, and 5% reactance. (b) The data for the photovoltaic (PV) system are given in Table 5.17. Determine the following: (i) The number of modules in a string, number of strings, number of arrays, surface area, weight, and cost. (ii) The inverter specification and the one-line diagram.

Solution A three-phase inverter rated at AC voltage 220 V can process approximately 20 kW. Based on the 20 kW three-phase inverter and 220 V AC output and the modulation index of 0.9, the input DC voltage is as given by Equation 5.16: Vidc =

2 2 × 220 3×0 9

= 399 V

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

311

TABLE 5.36 Weight Per Unit Power PV Type 1 2 3 4

Surface Area of One Module (ft2)

Power Rating (W)

Weight per Unit Power (lb/W)

33.07 39.00 40.70 18.30

190 200 170 87

0.174 0.195 0.239 0.210

Table 5.36 tabulates the data for each of the four types of PV modules. Table 5.36 shows the PV module type 1 has the minimum weight per unit power—hence our choice. The string voltage, SV, of the PV system should be close to the rated inverter input voltage, Vidc. Using string voltage, SV, of 400, from Equation (5.13), we have NM =

399 ≈7 54 8

Using seven modules, string voltage, SV, as given by Equation (5.14) is SV = 7 × 54 8 = 384 V The string power, SP, from Equation (5.9) is SP = 7 × 190 = 1 33 kW For designing an array to generate 20 kW, the number of strings, NS, in an array is as given by Equation (5.15): NS =

20 = 15 1 33

The number of arrays, NA, needed for this design is given by Equation (5.17): NA =

2000 = 100 20

And the total number of PV modules from Equation (5.19), TNM, = 7 × 15 × 100 = 10,500. The total surface area, TS, needed by the type 1 PV module is computed from the area of one module and the total number of modules: TS =

10,500 × 34 6 × 51 9 = 130,940 ft2 = 3 acre 144

312

SOLAR ENERGY SYSTEMS

The total weight, TW, for the type 1 PV module can be computed in a similar manner as TW = 10, 500 × 33 07 = 347,235 lb The total cost for the type 1 PV module is the product of the number of modules and the cost of each module as given below: The total cost of the PV modules = 10, 500 × 870 = $9 14 million Table 5.37 presents the PV specifications for 2000 kW generating station. We will use one inverter for each array. Hence, the rating of each inverter will also be 20 kW. The number of inverters, NI, needed for a generation of 2000 kW as from Equation (5.18) is NI =

2000 = 100 20

For 2000 kW PV generating station, we need 100 inverters to operate in parallel. Other designs using higher rating inverters are also possible. For this design, the final input DC voltage of the inverter is specified by the string voltage as Vidc = Vstring = 384 V With nominal DC input, the voltage of the inverter is selected, and the amplitude modulation index, Ma, is given as Ma =

2 2 × 220 3 × 384

= 0 93 V

If the switching frequency is selected at 10 kHz, the frequency modulation index, Mf, is given as Mf =

fS 10,000 = 166 67 = 60 fe

TABLE 5.37 The Photovoltaic Specifications for 2000 kW Generating Station Number Number Total of Power Total of per Area of Weight Total Cost Modules Strings Number String per of Voltage String the PV of the of the PV per PV Array Arrays (V) (kW) (ft2) PV (lb) (million $) Type String 1

7

15

100

384

1.33

130,940 347,235

9.14

313

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

TABLE 5.38 Inverter Specifications Number of Inverters 100

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

384

20

220

0.93

166.67

Amplitude modulation index = 0.93

DC bus 1

AC bus

DC/AC 384 V

Local power grid

Three-phase 220/460 V inverter 20 kVA 5% reactance

460 V/13.2 kV 13.2 kV 2 MVA 10% reactance

AC bus

100 DC/AC 100 PV arrays: 7 modules per string, 15 strings

Three-phase inverter

220 V

2000 kW 460 V

20 kW each

Figure 5.39 The one-line diagram of the system of Example 5.8.

Table 5.38 presents the inverter specifications. The one-line diagram of the system is given in Figure 5.39. We will use 100 transformers of 20 kVA and 5% reactance, which are connected to the inverters to step the voltage up from 220 to 460 V. Finally, one transformer of 460/13.2 kV and 10% reactance of 2 MVA is used to connect the system to the local power grid. Again, students should recognize that many designs are possible. Each design specification has its limitation that must be taken into account. The above analysis can be altered as needed to satisfy any design requirements. Example 5.9 Design a microgrid of a photovoltaic (PV) system of 1000 kW that is connected to the local utility bus at 13.2 kV by a 220 V/13.2 kV, 500 kVA, 5% reactance transformer. A local load of 500 kW is supplied at 220 V. A battery storage system of 100 kWh is fed from an AC bus of a PV system using a bidirectional rectifier that is used 8 hours a day. Solution Figure 5.40 depicts the microgrid of Example 5.9. We can select a type 1 PV module that has the most minimum weight among the four PV types as shown

314

SOLAR ENERGY SYSTEMS

DC bus

220 V/13.2 kV 500 kVA AC bus X = 5%

1

Local power grid

DC/AC Three-phase inverter Amplitude modulation index = 0.93

13.2 kV 500 kW 20 V

AC bus 50 DC/AC Three-phase 220 V inverter 384 V 20 kW 50 PV arrays: each 15 strings, 7 modules per string

220/20 V 2 kVA X=3 %

Bidirectional rectifier 1.56 kW 1 each

Bidirectional rectifier 4

4 battery arrays: 3 strings, 3 batteries per string

36 V DC bus

Figure 5.40 The one-line diagram of the PV system for Example 5.9.

in Table 5.36. The number of modules can be specified based on the selection of string voltage, SV, as given by Equation (5.13): NM =

400 ≈7 54 8

If the NM is equal to 7, then the SV from Equation (5.14) is SV = 7 × 54 8 = 384 V The string power, SP, from Equation (5.9) is SP = 7 × 190 = 1 33 kW If we design based on 20 kW in an array, then the number of strings, NS, in an array from Equation (5.15) is NS =

20 = 15 1 33

We can compute the number of arrays, NA, from Equation (5.17) as NA =

1000 = 50 20

315

A STORAGE SYSTEM BASED ON A SINGLE-CELL BATTERY

The total number of PV modules, TNM, from Equation (5.19) is TNM = 7 × 15 × 50 = 5250 The surface area, TS, for type 1 PV modules is TS =

5250 × 34 6 × 51 9 = 65, 470 ft2 = 1 50 acre 144

Similarly, we can compute the total weight and total cost: Total weight = 5250 × 33 07 = 173, 618 lb Total cost = 5250 × 870 = $4 56 million Table 5.39 presents the PV specifications for 1000 kW generating station. The rating of each inverter should be the same as the power generated by each array—20 kW. Because the input DC of the inverter is 384 V DC and the output AC voltage of the inverter is 220 V AC, the amplitude modulation index, Ma, of the inverter is given by Ma =

2 2 × 220 3 × 384

= 0 93

The number of inverters, NI, for a 1000 kW PV system from Equation (5.18) is NI =

1000 = 50 20

Therefore, we need to connect 50 inverters, one for each array for generation of 1000 kW. Let us select a switching frequency of 5.04 kHz. Therefore, the frequency modulation index is given by Mf =

fS 5040 = 84 = 60 fe

TABLE 5.39 The Photovoltaic Specifications for 1000 kW Generating Station Number of Number Number String Voltage of Modules of Strings PV (V) Type per String per Array Arrays 1

7

15

50

384

Total Total Area of Weight of Total Cost the of the PV the PV (ft2) PV (lb) (Million $) 65,470

173,618

4.56

316

SOLAR ENERGY SYSTEMS

TABLE 5.40 Inverter Specifications Number of Inverters 50

Input Voltage, Vidc (V)

Power Rating (kW)

Output Voltage, VAC (V)

Amplitude Modulation Index, Ma

Frequency Modulation Index, Mf

384

20

220

0.93

84

Table 5.40 presents the inverter specifications. One design choice for storing 100 kWh of energy is to use class 6 batteries from Table 5.23. These batteries are rated at 255 Ah at 12 V. Using three batteries per string and three strings in each array, we will have a battery storage system consisting of nine batteries in an array. The string voltage, SV, for the battery storage system is given below: SV = 3 × 12 = 36 V The energy stored, ES, is ES = Amp

hours × Voltage

ES = 255 × 12 = 3 06 kWh The energy stored in an array, ESA, is given by ESA = Number of batteries × kWh of one battery ESA = 9 × 3 06 = 27 54 kWh The number of arrays, NA, needed to store 100 kWh is given by NA =

Total energy Energy in each array

NA =

100 ≈4 27 54

Table 5.41 presents the battery storage array specification for 100 kWh. If the battery storage system is to supply PV loads for 8 hours, the battery storage system should be designed to discharge to 50% of capacity. Therefore, the

TABLE 5.41 The Battery Storage Array Specification for 100 kWh Battery Class 6

Number of Batteries per String

Number of Strings per Array

Number of Arrays

String Voltage (V)

Energy Stored per Array (kWh)

3

3

4

36

27.54

317

THE ENERGY YIELD OF A PHOTOVOLTAIC MODULE AND THE ANGLE OF INCIDENCE

TABLE 5.42 Bidirectional Rectifier Specifications for Charging the Storage System Number of Bidirectional Rectifiers 4

Input Voltage (V) AC

Power Rating (kW)

Output Voltage (V) DC

Amplitude Modulation Index

20

1.56

36

0.9

power provided by the battery storage system is 50% of the storage system capacity. For this design, using 50 kWh, Storage kW =

50 = 6 25 kW 8

And each array needs to supply one-fourth of the total kW or 1.56 kW. A bidirectional rectifier can be used to charge and discharge the stored energy system from the AC bus of a PV generating station (see Example 5.9). One bidirectional rectifier is used for each array of the battery storage system. A bidirectional rectifier with an amplitude modulation index of 0.9 with the output DC voltage equal to that of the battery array is selected. The AC input of the rectifier for an output of 36 V, with a modulation index of 0.9, is given by VL − L = =

3 × Ma × Vodc 2 2 3 × 0 9 × 36 2 2

= 20 V C

One rectifier is connected to each of the battery arrays, and each rectifier is designed to be rated at the same power rating of the battery array. Table 5.42 presents the bidirectional rectifier specifications for charging the storage system. The total power rating of all the rectifiers together is = 4 × 1.56 = 6.24 kW. The bus voltage at the inverter output terminals is 220 V. One transformer of 220/20 V, 3% reactance, 2 kVA is used to step down the voltage from 220 to 20 V for each of the bidirectional rectifiers. A 500 kVA, 220 V/13.2 kV, 5% reactance transformer is used to connect the system with the local power grid.

5.13 THE ENERGY YIELD OF A PHOTOVOLTAIC MODULE AND THE ANGLE OF INCIDENCE The angle of inclination for a module concerning the position of the sun must be determined for estimating the energy yield of a PV module. The angle of inclination is defined as the position that a magnetic needle makes with the horizontal plane at any specific location. The magnetic inclination is 0 at the magnetic equator and 90 at each of the magnetic poles. The irradiance is

318

SOLAR ENERGY SYSTEMS

defined as the density of radiation incident on a given surface expressed in W/ m2 or W/ft2. The angle at which the rays of the sun reach a PV module changes as the Earth rotates. The PV energy yield at a location as a function the PV module inclination angle is given in Appendix C. 5.14 THE STATE OF PHOTOVOLTAIC GENERATION TECHNOLOGY In recent years, the shift toward the development and installation of PV sources of energy has resulted in an explosion of growth in the research, development, and manufacture of PV systems. Germany is leading installed PV capacity in Europe, followed by Spain. Importantly, in recent years, this growth has been fueled by an increase in grid-connected systems. Today, 35% of the share of grid-connected cumulative installed capacity is composed of grid-PV-connected centralized applications. This further highlights the growing relevance of PV systems in fulfilling the ever-increasing energy demands of the twenty-first century.1–4,6,7 Students are encouraged to read the up-to-date report on the cost and efficiency of PV module from the National Center for Photovoltaics at NREL at https://www.nrel.gov/pv/ national-center-for-photovoltaics.html. In this chapter, we studied solar energy systems, specifically the development and design of a PV system with PV microgrid modeling. We also learned how to estimate the energy yield of a PV module based on the angle of inclination for a PV string concerning the position of the sun. The irradiance was defined as the density of radiation incident on a given surface expressed in W/m2 or W/ft2. Finally, we developed an estimation method to construct a model for a PV module. PROBLEMS 5.1

Search the Internet and specify four PV modules. Give your design and list the design data in Table 5.43. TABLE 5.43 Voltage and Current Characteristics of a Typical PV Module Power (max) Voltage at maximum power point (MPP) Current at MPP Voc (open-circuit voltage) Isc (short-circuit current) Efficiency Cost List five operating temperatures for Voc vs. Isc Width Length Height Weight

PROBLEMS

319

TABLE 5.44 Photovoltaic Module Data for Problem 5.3 Power (max) Voltage at Maximum Power Point (MPP) Current at MPP Voc (open-circuit voltage) Isc (short-circuit current)

400 W 52.6 V 6.1 A 63.2 V 7.0 A

5.2

Search the Internet to find the voltage–current characteristic of four PV modules. Make a table of input impedances as current varies for each operating temperature. Develop a plot of input impedance as a function of PV load current for each operating temperature.

5.3

Design a microgrid of PV rated at 100 kW of power at 230 V AC using a PV module of Table 5.44. Determine the following: (i) Number of modules in a string for each PV type. (ii) Number of strings in an array for each PV type. (iii) Number of arrays. (iv) Inverter specifications. (v) One-line diagram of this system.

5.4

Design a microgrid of PV rated at 600 kW of power at 460 V AC using a PV module with the data given in Table 5.43. Determine the following: (i) Number of modules in a string for each PV type. (ii) Number of strings in an array for each PV type. (iii) Number of arrays. (iv) Inverter specifications. (v) One-line diagram of this system.

5.5

Search the Internet for four single-phase inverters and summarize the operating conditions in a table and discuss the results.

5.6

Search the Internet for DC/DC boost converters and DC/AC inverters and create a table summarizing the operating conditions of four DC/DC boost converters and DC/AC inverters in a table and discuss the results and operations.

5.7

Design a microgrid of 50 kW, rated at 230 V AC. Use the PV module of Problem 5.3 and the converters of Problem 5.5. The design should use the least number of converters and inverters. Determine the following: (i) Number of modules in a string for each PV type. (ii) Number of strings in an array for each PV type.

320

SOLAR ENERGY SYSTEMS

(iii) Number of arrays. (iv) Converter and inverter specifications. (v) One-line diagram of this system. 5.8

Design a microgrid of 600 kW of power rated at 230 V AC. Use the PV module of Problem 5.3. The design should use the least number of converters and inverters. Determine the following: (i) Number of modules in a string for each PV type. (ii) Number of strings in an array for each PV type. (iii) Number of arrays. (iv) Converter and inverter specifications. (v) One-line diagram of this system.

5.9

Design a microgrid of a PV system rated at 2 MW and connected through smart net metering to the local utility at 13.2 kV. The local loads consist of 100 kW of lighting loads rated at 120 V and 500 kW of AC load rated at 220 V. The system has a 700 kWh storage system. Local transformer specifications are 13.2 kV/460 V, 2 MVA, and 10% reactance; 460 V/230 V, 250 kVA, and 7% reactance; and 460 V/120 V, 150 kVA, and 5% reactance. The data for this problem is given in Table 5.45. (i) Search the Internet for four DC/DC boost converters, rectifiers, and inverters and create a table. Summarize the operating conditions in a table and discuss the results and operations as relates to this design problem. Develop a MATLAB testbed to perform the following: (ii) Select boost converter, bidirectional rectifier, and inverters for the design of a microgrid from commercially available converters. If commercial converters are not available, specify the data for a new design of a boost converter, bidirectional rectifier, and inverters.

TABLE 5.45 Photovoltaic Module Data Panel

Type 1

Type 2

Type 3

Type 4

Power (max), W Voltage at max. power point (MPP), V Current at MPP, A Voc (open-circuit voltage), V Isc (short-circuit current), A Efficiency, % Cost, $ Width, in. Length, in. Thickness, in. Weight, lb

190 54.8 3.47 67.5 3.75 16.40 870.00 34.6 51.9 1.8 33.07

200 26.3 7.6 32.9 8.1 13.10 695.00 38.6 58.5 1.4 39

170 28.7 5.93 35.8 6.62 16.80 550.00 38.3 63.8 1.56 40.7

87 17.4 5.02 21.7 5.34 >16 397.00 25.7 39.6 2.3 18.3

PROBLEMS

321

(iii) Give the one-line diagram of your design. Make tables and give the number of modules in a string for each PV type, number of strings in an array for each PV type, number of arrays, number of converters, weight, and surface area required for each PV module type. (iv) Design a 700 kWh storage system. Search online and select a deepcycle battery storage system. Give the step in your design and include the dimension and weight of the storage system. (v) Develop a per unit model of the PV microgrid system. 5.10 Design a PV microgrid system operating at a voltage of 400 V DC serving a load of 50 kW and at 220 V AC. Use the datasets given in Tables 5.41 and 5.45. Perform the following: (i) Select a deep-cycle battery to store 100 kWh. (ii) Select a boost converter, bidirectional rectifier, and inverters for the design of a microgrid from commercially available converters and use the data in Tables 5.46–5.49 as applicable. If commercial converters are not available, specify the data for the new design of a boost, bidirectional rectifier, and inverters. (iii) Give the one-line diagram of your design. Make tables and give the number of modules in a string for each PV type, number of strings in an array for each PV type, number of arrays, number of converters, weight, and surface area required for each PV module type. 5.11

Write a MATLAB testbed for design PV system with minimum weight and minimum number of inverters using the data of Tables 5.41–5.48. Perform the following: (i) PV system for 5000 kW at 3.2 kV AC: Specify the inverter operating condition. (ii) PV system for 500 kW at 460 V AC: Specify the inverter operating condition. (iii) PV system for 50 kW at 120 V AC: Specify the inverter operating condition.

5.12 Consider the residential home of Figure 5.41. Perform the following: (i) Estimate the load consumption of the house. (ii) Plot the daily load cycle operation of the house’s loads over 24 hours and calculate the total energy consumption. (iii) Search the Internet and select a PV module and design the PV array for the house. Compute cost, the weight of the PV array, and roof areas needed for the PV system. Search the Internet and select an inverter, battery storage, and a bidirectional converter. 5.13

For Problem 5.11, if only 25% of the load is operated during the night, use the data of Problem 5.10 and specify a battery storage system to store the required energy for operating 25% of the load during the night.

TABLE 5.46 Typical Deep-Cycle Battery Data Overall Dimensions Part Number PVX-340T PVX-420T PVX-490T PVX-560T PVX-690T PVX-840T PVX-1080T PVX-1040T PVX-890T

322

Volts

Length (mm)

Weight (mm)

12 12 12 12 12 12 12 12 12

7.71 (196) 7.71 (196) 8.99 (228) 8.99 (228) 10.22 (260) 10.22 (260) 12.90 (328) 12.03 (306) 12.90 (328)

5.18 (132) 5.18 (132) 5.45 (138) 5.45 (138) 6.60 (168) 6.60 (168) 6.75 (172) 6.77 (172) 6.75 (172)

Capacity Ampere-Hours Height (mm)

6.89 8.05 8.82 8.82 8.93 8.93 8.96 8.93 8.96

(175) (204) (224) (224) (277) (277) (228) (227) (228)

Unit Wt lb (kg)

1H Rate

2H Rate

4H Rate

8H Rate

24 H Rate

48 H Rate

72 H Rate

120H Rate

25 (11.4) 30 (13.6) 36 (16.4) 40 (18.2) 51 (23.2) 57 (25.9) 70 (31.8) 66 (30.0) 62 (28.2)

21 26 31 36 42 52 68 65 55

27 33 39 45 53 66 86 82 70

28 34 40 46 55 68 88 85 72

30 36 43 49 60 74 97 93 79

34 42 49 56 69 84 108 104 89

36 43 52 60 73 90 118 112 95

37 43 53 62 76 94 122 116 98

38 45 55 63 79 97 126 120 102

PROBLEMS

323

TABLE 5.47 Boost Converters Input Voltage (V)

Output Voltage (V)

Power (kW)

26–48 26–63 26–80 26–80 26–80 26–100 82–160 82–200 82–300

9.2 12.2 11.23 11.23 13.1 12.5 15.2 14.2 9.5

24–46 24–61 24–78 24–78 24–78 24–98 80–158 80–198 80–298

TABLE 5.48 Single-Phase Inverter Data Inverter

Type 1

Type 2

Type 3

Type 4

Power Input voltage DC Output voltage AC Efficiency

500 W 500 V

5 kW 500 V max

15 kW 500 V

4.7 kW 500 V

230 VAC/60 Hz at 2.17 A

230 VAC/60 Hz 220 VAC/60 Hz 230 VAC/60 Hz at 27 A at 68 A at 17.4 A

Min. 78% at full load 15.5 5 5.3 9 lb

97.60%

>94%

96%

315 mm 540 mm 191 mm 23 lb

625 mm 340 mm 720 mm 170 kg

550 mm 300 mm 130 mm 20 lb

Length Width Height Weight

TABLE 5.49 Three-Phase Inverter Data Inverter

Type 1

Type 2

Type 3

Type 4

Power Input voltage DC Output voltage AC Efficiency

100 kW 900 V

250 kW 900 V max

500 kW 900 V

1 MW 900 V

Depth Width Height Weight

660 VAC/60 Hz 660 VAC/60 Hz 480 VAC/60 Hz 480 VAC/60 Hz Peak efficiency 96.7% 30.84 57 80 2,350 lb

Peak efficiency 97.0% 38.2 115.1 89.2 2,350 lb

Peak efficiency 97.6% 43.1 138.8 92.6 5,900 lb

Peak efficiency 96.0% 71.3 138.6 92.5 12,000 lb

324

SOLAR ENERGY SYSTEMS

PV arr a bo y x

TV DC bus

con Air ditio nin

g

Lig

DC hti

AC

Inverter

ng

Was he

r

Drye

Storage

r

Net smart meter

er

form

s Tran

AC bus

AC DC Bidirect ional conver ter

Figure 5.41

EV

Figure for Problem 5.12.

5.14

If the price of kWh from a utility company is $0.3 for buying or selling energy, estimate the net operating cost or revenue for the house of Problem 5.12.

5.15

Design a PV system rated at 50 kW using a boost converter and a DC/ AC inverter. The system operates as a stand-alone and supports a water pumping system with a rated load voltage of 120 V AC. Use the data given in Problem 5.10.

5.16

Design a residential PV system. The load cycle is 10 kW from 11 pm till 8 am and 14 kW for the remaining 15 hours. Determine the following: (i) Total kWh energy consumption for 15 hours. (ii) What is the roof space needed to generate adequate kWh for 24 hours’ operation? (iii) Assume the maximum kWh to be used during the night is 40% of the total daily load. Search the Internet to select a battery storage system and compute the required energy for nightly operation. Give your design data.

5.17

Design a microgrid of PV system rated at one MW of power at 220 V, 60 Hz, with all the PV strings connected to the same DC bus. The transformer data are 220/460 V, 250 kVA, and 5% reactance and 460 V/13.2 kV of 1 MVA and 10% reactance. Use the data given in Tables 5.41–5.48. Determine the following: (i) Number of modules in a string for each PV type, number of strings in an array for each PV type, number of arrays, and surface area, weight, and cost for each PV type.

PROBLEMS

325

(ii) Boost converter and inverter specifications and the one-line diagram of this system. 5.18

Assume a sample value for the global daily irradiation, G = [1900, 2690, 4070, 5050, 6240, 7040, 6840, 6040, 5270, 3730, 2410, 1800] for 12 months of the year. Assume a reflectivity of 0.25. Perform the following: (i) Write a MATLAB M-file program to (a) compute the irradiation on different inclination angles, (b) tabulate the irradiance for each month at different inclination angles, (c) tabulate the overall irradiance per year for different inclination angles, and (d) find the optimum inclination angle for each month and a year. (ii) If the solar irradiance is 0.4 sun for 8 hours daily for this location, what is the roof space needed to capture 20 kW at an optimal angle? (iii) If the solar irradiance is 0.3 sun on the average over a year for 5 hours daily for this location, what total kW can be captured over 1500 ft2 at the optimum inclination angle?

5.19

Assume the global daily irradiation (G) for the city of Columbus solar irradiation data, G, on the horizontal surface is as follows: G = [1800, 2500, 3500, 4600, 5500, 6000, 5900, 5300, 4300, 3100, 1900, 1500] for 12 months of the year. The latitudinal location of Columbus is 40 . Assume a reflectivity of 0.25. Perform the following: (i) Write a MATLAB M-file to (a) compute the irradiation on different inclination angles, (b) tabulate the irradiance for each month at different inclination angles, (c) tabulate the overall irradiance per year for different inclination angles, and (d) find the optimum inclination angle for each month and a year. (ii) If the solar irradiance is 0.4 sun for 8 hours daily for this location what is the roof space needed to capture 50 kW at an optimum inclination angle? (iii) If the solar irradiance is 0.3 sun on the average over a year for 5 hours daily for this location, what total kWh that can be captured over 1500 ft2 at the optimum inclination angle?

5.20

For your city, search the Internet for solar irradiation data, G, on the horizontal surface and its latitudinal location. Perform the following: (i) Write a MATLAB M-file to (a) compute the irradiation on different inclination angle, (b) tabulate the irradiance for each month at different inclination angles, (c) tabulate the overall irradiance per year for different inclination angles, and (d) find the optimum inclination angle for each month and a year. (ii) If the solar irradiance is 0.3 sun on the average over a year for 5 hours daily for this location, what is total kW that can be captured over 1500 ft2 at the optimum inclination angle?

326

5.21

SOLAR ENERGY SYSTEMS

For a PV module given in Table 5.50, write a MATLAB simulation testbed using Gauss–Seidel iterative approximation and estimate the module parameters (use Reference 18 and Gauss–Seidel iterative approximation). TABLE 5.50 Data for Problem 5.21 a1 a2 a3 a4 a5

(Isc) (Voc) (VMPP) (IMPP) (nc)

3.87 A 42.1 V 33.7 V 3.56 A 72

REFERENCES 1. California Energy Commission. Energy quest, the energy story. Chapter 15: Solar energy. Available at www.energyquest.ca.gov/story. Accessed June 10, 2009. 2. Elmhurst College. Virtual Chembook. Energy from the Sun. Available at http:// www.elmhurst.edu/~chm/vchembook/320sunenergy.html. Accessed July 10, 2009. 3. Deutsche Gesellschaft fur Sonnenenergie. Ebook on the Web: Planning and Installing Photovoltaic Systems: A Guide for Installers, Architects and Engineers. Available at http://www.ebookee.net/Planning-and-Installing-Photovoltaic-Systems-A-Guidefor-Installers-Architects-and-Engineers_181296.html. Accessed July 10, 2009. 4. British Petroleum. BP solar. Available at http://www.bp.com/genericarticle.do? categoryId=3050421&contentId=7028816. Accessed July 20, 2009. 5. Markvart, T. and Castaner, L. (2003) Practical Handbook of Photovoltaics, Fundamentals, and Applications, Elsevier, Amsterdam. 6. Cleveland, C.J. (2006) The Encyclopedia of Earth. Mouchout, Auguste. Available at http://www.eoearth.org/article/Mouchout,_Auguste. Accessed November 9, 2010. 7. U.S. Department of Energy, Energy Information Administration. Official energy statistics from the US Government. Available at http://www.eia.doe.gov/. Accessed September 10, 2009. 8. Wikipedia. Augustin-Jean Fresnel. Available at http://en.wikipedia.org/. Accessed October 9, 2009. 9. Carlson, D.E. and Wronski, C.R. (1976) Amorphous silicon solar cells. Applied Physics Letters, 28, 671–673. 10. Georgia State University. The doping of semiconductors. Available at http://hyperphysics.phy-astr.gsu.edu/hbase/solids/dope.html. Accessed November 26, 2010. 11. Energie Solar. Homepage. Available at http://www.energiesolar.com/energie/ html/index.htm. Accessed November 26, 2010. 12. American Society for Testing and Materials (ASTM) Terrestrial. ASTM standards and digital library. Available at http://www.astm.org/DIGITAL_LIBRARY/ index.shtml. Accessed November 26, 2010. 13. U.S. Department of Energy, National Renewable Energy Laboratory. Available at http://www.nrel.gov/. Accessed October 10, 2010.

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14. Siemens. Photovoltaic power plants. Available at http://www.energy.siemens.com/ hq/en/power-generation/renewables/solar-power/photovoltaic-power-plants.htm. Accessed October 10, 2010. 15. Gow, J.A. and Manning, C.M. (1999) Development of a photovoltaic array model for use in power-electronics simulation studies in electric power application, in IEEE Proceedings, Vol. 146. IEEE, Piscataway, NJ, pp. 193–200. 16. Esram, T. and Chapman, P.L. (2007) Comparison of photovoltaic array maximum power point tracking techniques. IEEE Transactions on Energy Conversion, 22(2), 439–449. 17. Sera, D., Teodorescu, R., and Rodriguez, P. (2007) PV panel model based on datasheet values, in Proceedings of the IEEE International Symposium on Industrial Electronics, IEEE, Piscataway, NJ, pp. 2392–2396. 18. Keyhani, A. Model Identification of Photovoltaic Generating Stations Simulation Testbed M File Codes: Photovoltaic Generating Stations, A Simulation Testbed, M File Codes, Amazon.com. 19. Quaschning, V. Understanding renewable energy systems. Available at http:// theebooksbay.com/ebook/understanding-renewable-energy-systems/. Accessed December 20, 2009. 20. Nourai, A. (2002) Large-scale electricity storage technologies for energy management, in Proceedings of the Power Engineering Society Summer Meeting, Vol. 1, IEEE, Piscataway, NJ, pp. 310–315. 21. Song, C., Zhang, J., Sharif, H., and Alahmad, M. (2007). A novel design of adaptive reconfigurable multicell battery for power-aware embedded network sensing systems, in Proceedings of Globecom, IEEE, Piscataway, NJ, pp 1043–1047.

CHAPTER 6

MICROGRID WIND ENERGY SYSTEMS 6.1

INTRODUCTION

Historians estimate that wind energy has been utilized to sail ships since about 3200 BC.1 The first windmills were developed in Iran (Persia) for pumping water and grinding grain.1 Denmark developed the first wind turbine for electricity generation in 1891.2 According to the Danish Energy Agency (https://ens.dk/en), the installed wind turbines reached 6100 units. Denmark wind energy capacity has more than doubled in the past 15 years, “with today’s 5.3 GW wind capacity installed on land and water” (https://ens.dk/en). Students can find updated information from the Department of Energy (DOE) (https://www.eia.gov/energyexplained/) on renewable energy resources currently consumed in the United States. The National Renewable Energy Laboratory (NREL) of the DOE3 of the United States is dedicated to the research, development, commercialization, and deployment of renewable energy sources. The NREL provides wind data by US location and cost of wind energy found at https://www.nrel.gov/workingwithus/partnering-facilities.html. According to wind exchange energy at https://windexchange.energy.gov/mapsdata/321, the United States alone possesses more than 8000 GW of land-based wind resources suitable for harnessing and an extra 2000 GW of shallow offshore resources.

Design of Smart Power Grid Renewable Energy Systems, Third Edition. Ali Keyhani. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/smartpowergrid3e

328

WIND POWER

6.2

329

WIND POWER

Wind energy is captured from the mechanical power of wind and converted to electric power using the classical process of Faraday’s law of induction.4 Wind energy is as one of our most abundant resources. It is the fastest-growing renewable energy technology worldwide.3,5–12 Improved turbine and power converter designs have promoted a significant drop in wind energy generation cost, making it the least expensive source of electricity (https://en.wikipedia. org/wiki/Wind_power_by_country). Global wind movement is predicated on the Earth’s rotation and regional and seasonal variations of solar irradiance and heating. Local effects on wind include the differential heating of the land and the sea and topography such as mountains and valleys. We always describe wind by its speed and direction. The speed of the wind is determined by an anemometer, which measures the angular speed of rotation and translates it into its corresponding linear wind speed in meters per second or miles per hour. The average wind speed determines the wind energy potential at a particular site. Wind speed measurements are recorded for a 1-year period and then compared to a nearby site with available long-term data to forecast wind speed and the location’s potential to supply wind energy. Wind speeds presented in Table 6.1 are based on a Weibull’s13 value of 2.0. Figure 6.1 depicts a wind turbine system. A horizontal-axis wind turbine is made of the following subsystems: 1. Blades: Two or three blades are attached to the hub of the shaft (rotor) of the wind generator; they are made of high-density epoxy or fiberglass composites. Wind exerts a drag force perpendicular to the blades and produces lift forces on the blades that cause the rotor to turn. The blades’ cross section is designed to minimize drag forces and boost lift forces to increase turbine output power at various speeds. 2. Rotor: The rotor transfers the mechanical power of the wind and acts to power the generator.

TABLE 6.1 Wind Power Classification Wind Power Class 3 4 5 6 7

Resource Potential

Wind Power Density at 50 m (W/m2)

Wind Speed at 50 m (m/s)

Wind Speed at 50 m (mph)

Fair Good Excellent Outstanding Superb

300–400 400–500 500–600 600–800 800–1800

6.4–7.0 7.0–7.5 7.5–8.0 8.0–8.8 8.8–11.1

14.3–15.7 15.7–16.8 16.8–17.9 17.9–19.7 19.7–24.8

330

MICROGRID WIND ENERGY SYSTEMS

3

2 4 6 1

9 4

12

8 7

10

13 11 14

15

Figure 6.1 A wind turbine system (Photo from the National Renewable Energy Laboratory).3,5,6

3. Pitch control uses an electric motor or hydraulic mechanism. It is used to turn (or pitch) the blades to maximize the power capture of the turbine or to reduce the rotor’s rotational speed in high winds. 4. A rotor brake system is used to stop the rotor for maintenance. Some advanced turbines use hydraulic brakes for the cut-in and the cut-out wind speeds when turbine power output is either too low or too high. 5. Low-speed shafts are designed to transfer the mechanical power of the rotor at a speed of 30–60 rpm to the gearbox. 6. A gearbox is used to couple low-speed and high-speed shafts and step up the rotational speed to 1200–1600 rpm suitable for electric generators. Gearboxes have disadvantages such as noise, high cost, frictional losses, and maintenance requirements, which preclude their use in some turbine designs. 7. Induction and permanent magnet generators are used for wind turbines. 8. Wind controllers are used to regulating and controlling the turbine’s electrical and mechanical operation. 9. An anemometer is used to measure the wind speed and sends the measurements to the controller. 10. A weather vane is an instrument for showing the direction of the wind. The vane is used to measure wind direction and sends it to the controller, which in turn commands the yaw drive to aim the turbine nose cone in the proper orientation as the wind direction changes.

WIND TURBINE GENERATORS

331

11. A nacelle is used as a weatherproof streamlined enclosure for housing shafts, a generator, a controller, and rotor brakes. 12. A high-speed shaft is used that mechanically couples the gearbox and rotor of the electric generator. 13. A yaw drive is used to orient the nacelle and the rotor using the yaw motor or hydraulic mechanism. 14. A yaw motor is used to move the nacelle with its components. 15. A tower is used to raise the turbine and hold the rotor blades and the nacelle. Turbine towers are tubular with heights approximately equal to the rotor diameter; however, minimum height is 26 m to avoid turbulence. Appendix D gives a summary of estimating the mechanical power of the wind. 6.3

WIND TURBINE GENERATORS

Wind turbine generators (WTGs) are rapidly advancing in both technology and installed capacity.13 Conventional geared wind generator systems have dominated the wind market for many years. Wind turbine technologies are classified based on their speed characteristics. They are either at fixed or variable speed. The speed of a wind turbine is usually low. The classical WTGs are of two types: (1) wound rotor winding and (2) squirrel-cage induction. These systems use multistage gear systems coupled to a fixed-speed squirrel-cage induction generator (SCIG), which are directly connected to the power grid. Figure 6.2a–e depicts an induction machine type with its windings, equivalent circuit, a view of wound rotor winding, a cutaway view of an SCIG, and a view of an SCIG.12 The concentrated representations of stator and rotor windings are represented in Figure 6.2a. However, in practice, the windings of a stator and rotor are approximately distributed sinusoidal windings. Figure 6.2b depicts the equivalent circuit model of the machine. The axes of these windings are displaced by 120 . The sinusoidal distribution of stator windings is measured with the angle ϕs, and sinusoidal distribution of rotor windings is measured with the angle ϕr. The angle θr represents the rotor angle as it rotates around the air gap. Figure 6.2c depicts a view of wound rotor winding. Figure 6.2d depicts a cutaway view of an SCIG. Figure 6.2e depicts an open view of a squirrel-cage winding of an induction generator. When the stator windings are excited with balanced three-phase sinusoidal currents, each phase winding produces a pulsating sinusoidal vector field along the winding axis and pointing to where the field is maximum positive as shown in Figure 6.2a. The effect of the three-phase winding vector field distribution is equivalent to having a single sinusoidal distributed vector field. For a two-pole machine, if the stator winding is excited by a 60 Hz source, the synchronous speed of the two-pole vector field is 3600 rpm. Figure 6.3 depicts the schematic of an SCIG system as a microgrid. An electronic switch is used to provide a soft start to smooth the connection and disconnection of the generator to the grid by limiting the unwanted inrush

332

MICROGRID WIND ENERGY SYSTEMS

(a) bs-axis br-axis

as′ a′r

cs cr b′r

ar

ϕr bs

br

ωr

Tr ar - axis

Ts θ r

ϕs as-axis

cr′ cs′

bs′ as

cs-axis

3-Phase supply

(b) Vabs Vbcs

Rext

as

as′

ar

a′r

bs

bs′

br

b′r

Rext

V′abr

cs

cs′

cr

cr′

Rext

V′bcr

(c) (c)

Figure 6.2 Induction machine types. (a) Wound machine stator and rotor windings. (b) Equivalent circuit. (c) A view of wound rotor winding. (d) A cutaway view of a squirrel-cage induction generator (SCIG). (e) A view of an SCIG.12 Source: (c)–(e) is courtesy of ABB (http://www.abb.com/product/us/9AAC133417.aspx).

333

WIND TURBINE GENERATORS

(d)

(e)

Figure 6.2 (Continued)

Soft starter SCIG

Infinite bus Local utility

Gearbox Capacitor bank

Figure 6.3 The schematic of a squirrel-cage induction generator (SCIG) system microgrid.

334

MICROGRID WIND ENERGY SYSTEMS

1 Current

Voltage

0.8

Voltage and current

0.6 0.4 0.2 Zero crossing

0 –0.2 –0.4 –0.6 –0.8 –1 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Time

Figure 6.4 The zero crossing of an AC source supplying power at 0.86 power factor lagging.

current to about 1.6 times the nominal current. The circuit breaker can be automatically controlled by a microcontroller or manually controlled. A zero crossing is the instantaneous point at which there is no voltage (current) present. In an AC alternating wave, this normally occurs twice during each cycle as shown in Figure 6.4. Assuming the system operates at a lagging power factor, for example, at a lagging power factor of 0.86, the current then lags the voltage as shown in Figure 6.4. The soft-switch circuit breaker is controlled by a microcontroller or a digital signal processor. The soft-switch circuit breaker is closed at a zero crossing of voltage, and it is open at zero crossing of current. The soft switch is also protecting the mechanical parts of the turbines such as a gearbox and shaft against high forces. The soft switch uses controllable thyristors to connect the generator to the induction generator at zero crossing of the sinusoidal voltage of the generator 2 seconds after the induction machine operates above the synchronous speed.

6.4

THE MODELING OF INDUCTION MACHINES

In the three-phase round induction machines with wound windings on the rotor, the stator windings consist of three-phase windings. The rotor winding also consists of three-phase wound windings. For a squirrel-cage induction machine, the rotor of the machine does not have winding; instead, it has a cage that is normally constructed from aluminum.

THE MODELING OF INDUCTION MACHINES

335

For mathematical modeling of the machine’s steady-state model,14,15 it is assumed that the stator and rotor windings are balanced. In the case of a squirrel-cage induction machine, it is assumed the rotor is represented by an equivalent winding. Each phase of the stator winding is designed using the same wire size and occupies the same space on the stator of the machine. The same is true for rotor winding. To understand the basic relationship between the current flowing in the windings and the resulting field distribution, we need to recall the relationship of the field distribution in the machine stator and rotor and machine air gap. Let us first review the fundamentals of the field distribution in a rectangular structure of an inductor depicted in Figure 6.5. These discussions facilitate our understanding of the basic concept of inductance and induction, which is used later in this chapter. According to the fundamentals of electromagnetics, the field intensity can be expressed as the product of H in A/m times the mean length of the magnetic core equal to a number of turns, N, times the current, I, flowing in the winding as given below: Hl = NI

(6.1)

From the fundamental relationship, the field intensity, H, and flux density, B, in weber per square meter (Wb/m2) and the permeability, μ, of the core material are related as H=

B μ

(6.2)

where μ = μrμ0; μ0 = 4π × 10−7 (H/m) and μr is relative permeability. For example, the relative permeability is equal to one for air. For iron, the relative permeability is in the range of 26,000–360,000. Therefore, μ is the slope of the B = μH and H of the field winding. The flux line distribution in the rectangular structure of Figure 6.5 can be computed by multiplying the field density by the cross-sectional area of the core, A, expressed as

� v

ϕ

ϕI

�

N

� Figure 6.5 An inductor winding.

336

MICROGRID WIND ENERGY SYSTEMS

Φ = BA

(6.3)

The flux linkage Φ represents the flux lines that link the winding; they are expressed in weber. The flux leakage, Φl, represents the flux lines that are leaked into the surrounding medium and do not remain in the core structure. If we assume the leakage flux lines are zero, then we can express the product of field intensity, H, and the mean length of the core to be equal to magnetomotive force, mmf, as shown by Equation (6.4): Hl = mmf

(6.4)

Therefore, the magnetomotive force is the force of the number of turn times of the current flowing through the winding, NI, which creates the force that produces the magnetic flux that is the result of the field intensity H. By substituting for H in Equation (6.4), we obtain the flux flow in the stator structure, rotor structure, and the air gap as Φ

l = mmf μ A

(6.5)

l μ A

(6.6)

ℜ=

Equation (6.6) is defined as reluctance. The reluctance is analogous to resistance. It is a function of the dimension of the medium through which the flux flows: Φ ℜ = mmf

(6.7)

The reluctance is inversely proportional to inductance, which is directly proportional to the square of the number of turns. The inductance, L, represents the inductance of the inductor of Figure 6.5: L=

N2 N2A or L = μ l ℜ

(6.8)

To understand the permeability, μ, we need to understand the hysteresis phenomena. A hysteresis loop shows the magnetic characteristics of a ferromagnetic material under test. The relationship shows the relationship between the voltage applied to a winding with a core structure made from a ferromagnetic material and resulting current flow in the winding. The induced magnetic flux density (B) is calculated from the applied voltage and the magnetizing field intensity (H) from the current flow through the winding. It is often referred to as the B–H loop. An example of a hysteresis loop is shown in Figure 6.6. The loop is produced by measuring the magnetic flux of a ferromagnetic material while the applied voltage is changed.

THE MODELING OF INDUCTION MACHINES

B

Flux density (mT)

337

Saturation a

Remanence b

Coercivity H

–H c

f

Field strength in opposite direction Saturation loop

Field strength (A/m)

e

d Saturation in opposite direction

–B

Figure 6.6 A schematic of a hysteresis loop.

A ferromagnetic core that has not been previously magnetized will follow the dashed line as the applied voltage is increased, resulting in higher current and higher magnetic intensity H.16 The greater the applied excitation current to produce, H+, the stronger the magnetic field density, B+. For point a, depending on the type of magnetic material, however, a higher applied voltage will not produce an increased value in magnetic flux. At the point of magnetic saturation, the process is changed in the opposite direction as shown in Figure 6.6, and recorded voltage and current representing flux density and field intensity move from point a to point b. At this point, some residual flux remains in the material even after the applied voltage is reduced to zero. This point is called retentivity on the hysteresis loop and shows the residual magnetism (remanence) in the material. As the applied voltage direction is changed, the recorded flux density and field intensity move to point c. At this point, the flux is reduced to zero. This is called the point of coercivity on the hysteresis loop. When the voltage is applied in a negative direction, the material is saturated magnetically in the opposite direction and reaches point d. As the applied voltage is increased in a positive direction, again H crosses zero at some value of the applied voltage, and the hysteresis loop reaches the point “e.” As the applied voltage is increased further, H is increased back in the positive direction, and the flux density, B, returns to zero. If the applied voltage repeats the same cycle, the process is repeated. If the process is reversed, the loop does not return to original position. Some force is required to remove the residual magnetism and to demagnetize the material.16

MICROGRID WIND ENERGY SYSTEMS

Normal magnetization curve

Flux density (mT)

338

Remanence A

B C

Coercivity Field strength (A/m) Saturation loop

Figure 6.7 A family of hysteresis loops as applied voltage to the winding is varied.

Figure 6.7 shows the traces of a family of hysteresis loops as applied voltage to the winding is varied. When the tips of the hysteresis loops A, B, and C are connected, the result is a normalized magnetizing curve. Figure 6.8 depicts the normalized magnetizations curve. Because B = μH, therefore, the permeability, μ, is the slope of hysteresis as expressed by Equation (6.9): μ=

B H

(6.9)

In the linear region of the magnetization curve, the inductance is linear as given by Equation (6.8). Figure 6.8 shows the change in inductance of the rectangular core, as the core is saturated. Figure 6.9 depicts the schematic of inductance as a core structure saturated. Figure 6.5 represents a simple rectangular structure. However, a three-phase induction machine has three windings on the stator and three windings on the rotor as shown in Figure 6.2a. The three windings on the stator are placed on the circumference of the stators at 120 apart. The three-phase rotor windings are also located on the circumference of the rotor at 120 apart. As a motor operation, the stator is supplied from the three-phase AC voltage source.

THE MODELING OF INDUCTION MACHINES

339

B

H

Figure 6.8 The normalized magnetization curve.

L

i (or H)

Figure 6.9 Schematic of inductance as a core structure saturated.

The time-varying AC source would magnetize the stator structure and give rise to time-varying flux density and time-varying inductance for each phase of the stator. Although the cylindrical structure is more complex to calculate these inductances, we can use advanced finite element calculation and make discrete elements of the stator structure to calculate the flux in these elements. However, finite element calculation is an advanced topic that is not the subject of this book. From our study of the inductor, we can conclude that the inductances of the stator are a function of the diameter, a number of turns, the permeability of the stator material, and cross-sectional area and excitation current. Therefore, we recognize that the flux density variation in winding a-a when it is energized from the AC alternating source can be expressed as Ba θr , t = B t cos

P θr D

(6.10)

where B(t) is time-varying flux density produced by the phase “a” applied voltage, va(t): B t = Bmax cos ωs t

(6.11)

340

MICROGRID WIND ENERGY SYSTEMS

In Equation (6.11), the ωs = 2πf is the source frequency and θr is the angle around the circumference of the stator, P is a number of poles, D is diameter, and (P/D)θr is angle in radians. For example, Ba(x, t) = Bmax when θr = 0. And θr = 2πD/P, where Equation (6.10) can be expressed as Ba x, t = B t cos

P 2πD = Bmax D P

(6.12)

Substituting Equation (6.12) in Equation (6.10), we can obtain Equation (6.13): Ba x, t = Bcos ωs tcos

P θr D

(6.13)

Using the cosine identity in Equation (6.13), we can obtain Equation (6.14): 1 cos α −β + cos α + β 2 B Pθr B Pθr Ba x, t = cos − ωs t + cos + ωs t 2 2 D D

cos αcos β =

(6.14)

We can obtain the same expression for windings b-b and c-c as expressed by Equations (6.15) and (6.16): Bb x,t = Bcos ωs t −

2π Pθr 2π cos − 3 3 D

(6.15)

Bc x,t = Bcos ωs t −

4π Pθr 4π − cos 3 3 D

(6.16)

Recalling that the flux density is a vector quantity, we can add flux density generated by phase a, phase b, and phase c, and we can obtain the total flux generated as expressed by Equation (6.17): Btot x, t = Ba θr , t + Bb θr , t + Bc θr , t

(6.17)

3 Pθr − ωs t Btot x, t = B cos 2 D

(6.18)

The peak value of flux density crossing the air gap of the machine is given by Equation (6.19): 3 Bmax = B 2

(6.19)

THE MODELING OF INDUCTION MACHINES

341

The rotor windings are distributed on the rotor structure. The total timevarying flux density of stator windings crossing the air gap links the rotor windings and induces a voltage in the rotor windings. Let us analyze the rotor-induced voltage assuming two conditions: (1) the rotor is at a standstill, and (2) the rotor is free to rotate. 1. The rotor is at a standstill. Btot(θr, t) is total flux distribution in the machine and frequency of flux wave is expressed as ωs = 2π fs

(6.20)

where fs is the stator frequency. This flux distribution can be regarded as a flux wave that is distributed in the air gap machine with mechanical equivalent speed as ωmech equivalent =

2 ωs P

(6.21)

The mechanical speed expressed by Equation (6.21) is also called the synchronous speed: ωsyn =

2 ωs P

(6.22)

Let us assume that we short the rotor windings and assume that the rotor is restrained and remains at a standstill. The stator flux wave distribution of the stator crosses the machine air gap and links the rotor windings and induces the voltage in phases a, b, and c of the rotor windings. The resulting current flow in rotor windings gives rise to the rotor flux of rotor windings. Adding flux waves produced by phases a, b, and c of rotor windings, we can obtain BR x, t = BaR x, t + BbR x, t + BcR x, t 3 Pθr − ωs t = B cos D 2

(6.23)

Because the rotor is at a standstill, that shaft speed is zero; the rotor frequency is the same as stator frequency. The induction machine is acting like a three-phase transformer; however, in a cylindrical structure with the rotor supported on a bearing system that separates the stator from the rotor with a very small air gap, the stator and rotor windings are coupled.

342

MICROGRID WIND ENERGY SYSTEMS

2. The rotor is free to rotate. Suppose the rotor is free to rotate. Let us assume rotor is rotating at a speed of ωm. Then the rotor total flux wave distribution, BR(x, t), has the mechanical speed as expressed by Equation (6.24): ωr mech = ωsyn −ωm

(6.24)

Here ωr(mech) is the rotor mechanical speed that is the difference between the synchronous speed and the shaft speed because the shaft is free to rotate. The electrical speed of the rotor flux wave is expressed as given by Equation (6.25): ωr =

P P ωr mech = ωsyn − ωm 2 2

(6.25)

ωr = 2πfr where fr is the frequency of induced voltage (current) in the rotor. The total flux distribution of stator flux Bs(θr, t) has the equivalent mechanical speed at ωsyn. This flux distribution for a two-pole machine has two poles as shown in Figure 6.10. The total flux distribution of the rotor, BR(θr, t) of the rotor, rotates at ωsyn − ωm for a two-pole machine with respect to the rotor structure and BR(θr, t) of the rotor structure rotates at ωm (mechanical speed) with respect to the stator. The two flux waves are distributed in the machine at an angle with respect to each other as shown in Figure 6.10. Motor action ωsyn and ωm are rotating in the same direction and ωsyn > ωm.

S

S N ωm TL N

ωsyn

Tem

Figure 6.10 The schematic flux distributions for two-pole machines.

THE MODELING OF INDUCTION MACHINES

6.4.1

343

Calculation of Slip

ωr =

P ωsyn −ωm 2

(6.26)

Let us multiply and divide by ωsyn (stator electrical speed): ωr =

ωsyn − ωm ωsyn 2 ωsyn P

(6.27)

Let us define ωsyn −ωm = slip ωsyn

(6.28)

ωr = sωsyn or fr = sf

(6.29)

s=

where ωsyn rad/s, ωm rad/s. The slipcan be calculatedas presentedinEquation(6.30): s= nsyn =

nsyn − nm nsyn

(6.30)

120fs P

(6.31)

where nsyn is the synchronous speed in rpm and nm is the rotor speed in rpm.

6.4.2

The Equivalent Circuit of an Induction Machine

Figure 6.11 depicts a three-phase Y-connected stator and rotor windings. As a motor operation, the stator is fed from a three-phase AC source or through a pulse width modulation (PWM) inverter with a variable frequency. The rotor is short-circuited or connected to an external resistance. As a motor, c

b a Grid Rext

Vgrid

Stator ““1” Tmech

Tem

Rotor “2”

Figure 6.11 The equivalent circuit of a wound rotor induction machine with external resistance, Rext, inserted in the rotor circuit.

344

MICROGRID WIND ENERGY SYSTEMS

I1

R1

jωsL1

jωsL2

I′2

R2

I2

Rext

jXm Vgrid

Rm

E1

E2

N1

N2

Shaft ωm = 0

Figure 6.12 The equivalent circuit model of an induction machine at standstill.

the machine operates at a shaft speed below the synchronous frequency and supports a mechanical load. As a wind generator, the mechanical wind power is supplied, and the generator operates above the synchronous speed. To describe how the machine operates, we need to develop an equivalent circuit model for the machine. In Figure 6.12, we refer to the stator winding with subscript “1” and rotor as subscript “2.” We look at two operating conditions: (1) the rotor is at a standstill, and (2) the rotor is free to rotate. 1. The rotor is at a standstill. Suppose the rotor is at standstill and the shaft speed is zero (ωm = 0). Based on the modeling the coupled winding, the equivalent circuit for one phase to the ground can be depicted in Figure 6.12. The machine is designed to use a small amount of current to magnetize it. Normally, the magnetizing current is less than 5% of the rated load current. This requires that magnetizing reactance, Xm, and resistance, Rm, which is net magnetizing impedance, acts as high impedance. Therefore, we can ignore the shunt elements. With this assumption, we can obtain the following: I 1 = I2

(6.32)

V grid = I 1 R1 + jωs L1 + E1

(6.33)

E2 = I 2 R2 + Rext + jωs L2

(6.34)

2. The rotor is free to rotate. Suppose rotor speed is running at ωm. Let us express R2 = R2,rotor + Rext. Figure 6.13 depicts the equivalent circuit model of an induction machine when the rotor is free to rotate:

THE MODELING OF INDUCTION MACHINES

I1

jωsL1

R1

jωrL2

Vgrid

E1

E2

N1

N2

R2

345

I2

Tem TL

Figure 6.13 The equivalent circuit model of an induction machine when the rotor is free to rotate.

V grid = I 1 R1 + jX 1 + E1

(6.35)

E2 = I 2 R2 + jωr L2

(6.36)

where R1 is stator resistance/phase and X1 = ωsL1 is stator reactance/ phase, I1 is stator current (line), and E1 is induced emf/phase. Let us define a=

N1 N2

(6.37)

I1 =

I2 a

(6.38)

Based on Faraday’s law of induction, the following expression defines the induced voltages, e1 and e2: e1 = N1

dΦ = N1 Φmax cos ωt + θ1 dt

(6.39)

e2 = N2

dΦ = sN 2 Φmax cos ωt + θ2 dt

(6.40)

Then, the root mean square (RMS) values E1 and E2 can be expressed as E1 =

N1 Φmax 2

and E2 =

sN 2 Φmax 2

(6.41)

The ratio of E1 and E2 can be computed as E2 = sN 2

E1 E1 =s N1 a

(6.42)

346

MICROGRID WIND ENERGY SYSTEMS

And I2 and I1 can be expressed as I 2 = aI 1

(6.43)

Using the above relationship, we can compute the equivalent model of induction machine from the stator side by referring the rotor variables to the stator sides: E2 = I 2 R2 + jωr L2 E1 s = aI 1 R2 + jωr L2 a

(6.44) (6.45)

Equation (6.45) can be recomputed by multiplying by a and dividing by s: sE1 = I 1 a2 R2 + jsa2 X2

(6.46)

R2 + ja2 X2 s

(6.47)

E1 = I 1 a2

Equation (6.47) represents the rotor variables from the stator side. The one-phase stator variables are given by Equation (6.48): V grid = I 1 R1 + jX 1 + E1

(6.48)

Combining Equation (6.47) with Equation (6.48), we obtain the following: V grid = I 1 R1 + jX 1 + I 1 a2

R2 + ja2 X2 s

(6.49)

We can rewrite Equation (6.49) as defined by Equation (6.50): R2 = a2 R2 X2 = a2 X2 V grid = I 1

R1 +

R2 + j X1 + X2 s

(6.50) (6.51)

The induction motor equivalent circuit from the stator side is given in Figures 6.14 and 6.15. 6.5

POWER FLOW ANALYSIS OF AN INDUCTION MACHINE

The input power can be calculated from the input voltage and current drawn by the machine as given by Equation (6.52):

POWER FLOW ANALYSIS OF AN INDUCTION MACHINE

jX1

R1

P, Q +

jX2

aeff +

I1 RC Core loss

Vgrid –

347

+I2

jXmE1

R2

E2

–

– TL

Tem

Figure 6.14 The equivalent circuit model of an induction machine with magnetizing inductance represented on the stator side.

+ Vgrid –

jX1

R1

P,Q

R′2

jX′2 I2

I1 Core loss

RC

R′2 (1 – s) s

jXm

Tem

TL

Figure 6.15 The equivalent circuit model of an induction machine with rotor variables referred to the stator side.

Pi = 3Re V1 I1∗

(6.52)

The power crossing the air gap of the machine (see Figure 6.14 or 6.15) can be calculated by subtracting stator wire resistance losses and core losses as given by Equation (6.53): PAG = 3Pi − 3 I12 R1 + Pc = 3 I2

2

R2 s

(6.53)

where Pc is the core loss shown by equivalent resistance Rc in Figure 6.15. As it can be observed from Equation (6.53), the air gap power can also be R computed by the square of the rotor current I2 multiplied by 2 . s When the machine is operating as a motor, the power delivered to the shaft can be calculated by accounting for rotor losses: Pem = PAG − 3I22 R2

(6.54)

By substituting for the air gap power in Equation (6.54), we can obtain Equation (6.55):

348

MICROGRID WIND ENERGY SYSTEMS

Pem = 3I

2 2

R2 2 − 3I 2 R2 = 3I s 2

Pconv = Pem = 3I 2 R2

2 2

R2 −R2 s

(6.55)

1 −s s

(6.56)

Therefore, the electromagnetic power delivered to the machine shaft can be expressed as a function of air gap power by Equation (6.57): Pem = 3I

2 2

R2 1 − s = PAG 1− s s

(6.57)

The flow of power from the stator to the machine shaft is depicted by a power flow of Figure 6.16. The power flow is given from input power, Pi, the air gap power, PAG, electromagnetic power, Pem, and output power, Po, to the shaft of the machine. An induction machine has three regions of operations. It can operate as a motor, as a generator, or as a break. To describe these regions of operation, we need to study the machine torque as a function of speed. Figure 6.17 depicts the equivalent circuit model of an induction machine, operating as a motor, from stator terminals omitting the magnetizing elements. In Equation (6.58), ωm is the shaft speed, Tem is torque driving the shaft, and Pem is the power supplied to the machine from the electrical supply: Pem = Tem ωm

(6.58)

We can calculate Pem by calculating the current supplied to the machine from the grid. Because the magnetizing element is ignored, the current I1 is equal to I2 : Vgrid = I1

R 1 + R 2 + j X 1 + X2

(6.59)

From the above equation, we can calculate the current I1 as expressed by Equation (6.60):

PAG = Pi – PR1

Pi

PR1 = I12 . R1

Pem = PAG(1 – s)

PR2 = I22 . R′2

Pem

Rotational losses

Figure 6.16 The power flow in induction machines.

POWER FLOW ANALYSIS OF AN INDUCTION MACHINE

+

jX1

R1

P,Q

R′2

349

jX′2 I′2

I1

R′2

Vgrid

(1 – s) s

– Tem

TL

Figure 6.17 The equivalent circuit model of an induction machine, operating as a motor, from stator terminals omitting the magnetizing elements.

I1 2 =

Vgrid R R1 + 2 s

2

(6.60)

2

+ X1 + X 2

2

The torque supplied to the rotor is expressed by Equation (6.61): Tem =

3 1 −s R2 I 1 ωm s

2

(6.61)

The shaft speed can also be expressed as given by Equation (6.62): ωm = ωsyn 1− s

(6.62)

By substituting Equations (6.61) and (6.62), we can express the shaft torque as a function of the input voltage as given by Equation (6.63): Tem =

3 R2 ωsyn s

Vgrid R R1 + 2 s

2

2

+ X1 + X2

(6.63) 2

It is instructive to study the machine performance for various values of external resistance for a constant input voltage. From a careful examination of Equation (6.63), we can conclude the following: a. When the slip is zero, the shaft speed, ωm, is equal to the synchronous speed, ωsyn, and the produced electromagnetic torque is zero. b. When the slip, s, is equal to one, that is, the shaft speed is zero (starting), the standstill torque (starting torque) can be obtained. c. As the external resistance is changed using a controller, the value of maximum torque occurs at different values of the shaft speed. The control of shaft speed over a wide range would provide the capability to operate the machine at various speeds. When the machine is used as an

350

MICROGRID WIND ENERGY SYSTEMS

Motor

Torque

Braking region

Motor region Generator

Generator region

–100 – 80 – 60 – 40 – 20 0 20 40 60 80 20 120 140 160 180 200 220 Speed in percent of synchronous speed 2.0 1.8

1.6

1.4

1.2

Figure 6.18

1.0 0.8 0.6 0.4 0.2 0 – 0.2 – 0.4 – 0.6 – 0.8 – 1.0 – 1.2 Slip as a fraction of synchronous speed

An induction machine’s various regions of operation.

induction generator, we can capture the wind power and inject the generated power into the local grid. There are three distinct points in an induction machine torque versus speed curve: (1) the starting torque when the shaft speed is zero; (2) when the shaft speed is equal the synchronous speed of the flux waves, the generated torque is zero (see Figure 6.18); and (3) the point of maximum torque production. To calculate the maximum torque, we compute the derivative of torque expression as given by Equation (6.63) with respect to slip and set it to zero: dTm =0 ds

(6.64)

The maximum slip point is given by Equation (6.65): R2

smax = ± R21

(6.65)

+ X1 + X 2

2

The resulting maximum torque is given by Equation (6.66): Tmax = ±

3 2ωsyn

Vgrid R1 ±

R21

2

+ X1 + X2

(6.66) 2

THE OPERATION OF AN INDUCTION GENERATOR

351

Equation (6.65) shows that the slip at which the maximum torque occurs is directly proportional to the rotor resistance. Equation (6.66) shows that the maximum torque is independent of the rotor resistance. However, it is directly proportional to the square of the input voltage. 6.6

THE OPERATION OF AN INDUCTION GENERATOR

Wound rotor induction machines have stators like the squirrel-cage machine. However, their rotors’ winding terminals are brought out via slip rings and brushes for torque and speed control. For torque control, no power is applied to the slip rings. External resistances are placed in series with the rotor windings during starting to limit the starting current. Without the external resistances, the starting currents are many times the rated currents. Depending on the size of the machine, it can draw 300% to over 900% of full-load current. The resistances are shorted out once a motor is started. The external resistances are also used to control the machine speed and its starting torque. The negative slip operation of an induction machine indicates that the machine is operating as a generator and power is injected into the local power grid. Induction generators required reactive power for magnetizing the rotor, and this power is supplied by the following different methods: 1. The machine is magnetized and started as an induction. Then, the wind turbine is engaged to supply mechanical power, and shaft speed is increased above synchronous speed. As a stand-alone, we can use any induction machine, and by adding capacitors in parallel with the machine terminals and then driving it above synchronous speed, the machine would operate as an induction generator. The capacitance helps to induce currents into the rotor conductors. The loads can be connected to the capacitor leads because the capacitors are in parallel. This method of starting a stand-alone machine is assured if the residual magnetism in the rotor exits. However, the machine can be magnetized by using a direct current (DC) source such as a 12 V battery for a very short time. 2. Rotor resistance of the induction generator is varied instantly using a fast power electronics controller. Variable rotor speed (consequently variable slip) provides the capability to increase the power captured from the wind at different wind speeds. This can be achieved if rotor winding terminals can be accessed by changing slip via external resistors connected to the generator rotor winding. The external resistors will only be connected to produce the desired slip when the load on the wind turbine becomes high. 3. Figure 6.19 depicts (a) equivalent circuit of an induction generator, (b) the equivalent circuit of an induction generator with all quantities referred to the stator side, and (c) the equivalent circuit of an induction generator referred to the stator neglecting magnetizing and loss components.

352

MICROGRID WIND ENERGY SYSTEMS

(a)

P

+

Q

IG

jX1

R1

jX2

aeff +

Im RC Core loss

Vgrid –

+

jXmE1 – Tem

R2

E2 – TL

(b) Q

P +

IG,1

Im,1 RC Core loss

Vgrid –

(c)

P

+

jX1

R1

IG

Q

R1 Im

jXm

R′2

jX′2

IG,2

Im,2 + Vind –

jX1

R′2

–

Tem

TL

R′2

(1 – s) s

jX′2 + Vind

Vgrid

R′2 (1 – s) s

– Tem

TL

Figure 6.19 (a) Equivalent circuit of an induction generator. (b) Equivalent circuit of an induction generator with all quantities referred to the stator side. (c) Equivalent circuit of an induction generator referred to the stator neglecting magnetizing and loss components.

The relationships between the variation of external resistance and the location of maximum and standstill torques can be studied by varying the external resistance in the rotor circuit. We can study this relationship by writing a MATLAB M-file as described in Example 6.1. The power from the rotation of the wind turbine rotor is transferred to the induction generator through a transmission train consisting of the main turbine shaft, the gearbox, and the high-speed generator shaft. The normal range of wind speed is not high and changes during the day and with the seasons. For example, a three-phase generator with two poles directly connected to a local power grid operating at 60 Hz is running at 3600 rpm. We can reduce the speed of the generator rotor if the generator has a higher number of poles. For example, if the number of poles is four, six, or eight, the shaft speed reduces to 1800, 1200, and 900 rpm, respectively. A power grid supplied by

THE OPERATION OF AN INDUCTION GENERATOR

353

a wind generator requires much higher wind speed in the range of 900–3600 rpm. To reduce the speed of the generator rotor, we can increase the number of poles. However, designing a generator with a very high number of poles requires a large diameter and will result in high volume and weight. We can increase the low speed and high torque of the wind turbine to low torque and high speed using gear systems. Before discussing the gear concepts, we need to understand how to convert linear velocity to rotational velocity. From fundamental physics, we know the linear velocity and rotational velocity can be expressed as V =r ω

(6.67)

where V is the linear velocity in meters per second, r is the radius in meter, and rotational velocity ω is in radians per second. We can convert the rotational velocity, ω, to revolutions per minute as ω=N

2π 60

In the above, N is in revolutions per minute. Equation (6.67) can be restated as V =N

2π r 60

(6.68)

where V is in meters per second, N is in rotations per minute, and r is in meters. Conversely, we can rewrite the above as N =V

60 2π r

(6.69)

In Equation (6.69), V is in meters per second, r is measured in meter, and N is in rotations per minute. Conversely, the unit in Equation (6.69) can be in feet per second for V, r is in feet, and N is in rotations per minute. If we want to express the velocity in miles per hour (mph), we rewrite Equation (6.69) as 5280 πr N V= 3600 30

(6.70)

where V is in miles per hour, r is in feet, and N is in rotations per minute. In Equation (6.70), we have substituted the value of 5280 feet for one mile and 3600 seconds for an hour. Equation (6.70) can be restated as N=

14 01 V r

(6.71)

354

MICROGRID WIND ENERGY SYSTEMS

4 to 1 gear ratio

24 6

4 turns rotates 24 teeth

1 turn rotates 24 teeth

Figure 6.20 A schematic of a gear mechanism.

The wind speed in the range of 10 miles per hour (14.67 ft/s) will result in 124.2 rotations per minute. The wind generators are designed to capture the wind power in the range less than 120 rpm using a gearbox transmission system. The gearbox operates as a transformer. The gearbox provides speed and torque conversions. Figure 6.20 depicts a schematic of a gear mechanism: Tinput ωinput = Toutput ωoutput

(6.72)

where Tinput and Toutput are torque in newton-meter and ωinput and ωoutput in radian per second. For torque–speed conversion, a gearbox transmission is used. A gearbox is designed with a number of teeth: Gear ratio =

Tinput Input number of teeth = Output number of teeth Toutput

(6.73)

The gearbox converts low speed and high torque powers of a wind turbine rotor to high speed and low torque powers of the generator rotor. The gear ratio is in the range of 1–100 for a wind generator in the range of 600 kW to 1.5 MW. Example 6.1 Consider a three-phase Y-wound rotor-connected induction machine operating as a generator in parallel with a local power grid. The machine is rated at 220 V, 60 Hz, and 14 kW, with eight poles and the following parameters: Stator resistance (R1) of 0.2 ohms per phase (Ω/phase) and reactance of (X1) of 0.8 Ω/phase.

THE OPERATION OF AN INDUCTION GENERATOR

355

Rotor resistance (R2 ) of 0.13 Ω/phase and reactance (X2) of 0.8 Ω/phase. Ignore magnetizing reactance and core losses. Perform the following: (i) Develop a one-line diagram and one-phase equivalent model. (ii) If the prime mover speed is 1000 rpm, determine the active and reactive power between the local grid and wind generator. How many capacitors must be placed at the machine stator terminal for unity power factor operation? (iii) Plot the torque–speed characteristics of the machine at different values of external resistances. Solution The synchronous speed is Ns =

120f 120 × 60 = = 900 rpm P 8

The rotor speed is Nr = 1000 rpm The slip is s=

Ns −Nr 900 −1000 = − 0 111 = 900 Ns

Writing Kirchhoff’s voltage law (KVL) for the circuit of Figure 6.21b, the current in the motor convention is Im = =

Vgrid R 1 + R 2 s + j X 1 + X2 220 3 = 67 88 ∠ −121 22 0 2 −0 13 0 111 + j 0 8 + 0 8

For motor operation, the power flows from the grid to the induction machine. For generator operation, the direction of active power is reversed. Using motor convention and calculating the current, the angle of the current is greater than 90 . This means that the direction of the current is opposite to the direction as shown by the generator convention in Figure 6.21b.

356

MICROGRID WIND ENERGY SYSTEMS

(a) Induction generator Gearbox

Local power grid

(b) Q

P

R1

+

IG

jX1

Im

R′2

jX′2 + R′2 (1 – s) s

Vind

Vgrid

–

–

Tem

TL

Figure 6.21 (a) A one-line diagram. (b) A one-phase equivalent circuit of an induction generator.

With current following the generator convention and flowing from the induction generator to the grid, the current is IG = 67 88 ∠ 180 −121 22 = 67 88 ∠ 58 77 The induced voltage is given as Vind = =

1−s R IG s 2 1 − − 0 111 × 0 13 × 67 88 ∠ 58 77 = 88 25 ∠ 58 77 − 0 111

Neglecting the mechanical losses, the electrical power generated from the supplied wind mechanical power is given as ∗ = 3 × 88 25 ∠ 58 77 × 67 88 ∠ − 58 77 Sind = 3Vind IG

= 17 93 + j0 kVA From Sind, it is seen that the induction generator produces active power. However, it does not produce any reactive power. The reactive power is supplied by the grid.

THE OPERATION OF AN INDUCTION GENERATOR

357

Induction generator Gearbox

Local power grid

Cp

Figure 6.22 A squirrel-cage induction generator with reactive power supplied locally by a capacitor bank.

The power produced by the induction generator is injected into the local grid, and some power is lost in stator and rotor resistance. The complex power injected into the grid is given by ∗ Sgrid = 3Vgrid IG

=3×

220 3

∠ 0 × 67 88 ∠ −58 77 = 13 41− j22 12 kVA

The induction generator feeds 13.41 kW of active power to the grid but consumes 22.12 kVAr of reactive power from the grid. Let a three-phase Y-connected capacitor bank be connected at the terminals of the induction generator. For unity power factor operation, the capacitor bank must supply the reactive power demand of the induction generator: CP =

Q 22 12 × 103 = 2 2πf 3Vgrid 2π × 60 × 3 × 220

3

2

= 1 2 mF

Figure 6.22 depicts a squirrel-cage induction generator with reactive power supplied locally by a capacitor bank. Let us assume that the value of external resistance is varied from 0 to 0.75 in steps of 0.25 and the speed of the machine is varied from zero to the synchronous speed. In the following MATLAB testbed, a plot is made for the different values of Rext. %TORQUE vs SPEED clc; clear all; v1=220/sqrt(3); f=60; P=8; r1=0.2; x1=0.8;

358

MICROGRID WIND ENERGY SYSTEMS

r2d=0.13; x2d=0.8; % The electrical quantities are defined ws=120*f/P; Tmax=-(3/2/ws)*v1^2/(r1+sqrt(r1^2+(x1+x2d)^2)) Tmax_gen=-(3/2/ws)*v1^2/(r1-sqrt(r1^2+(x1+x2d)^2)) w=0:1:2*ws; for r_ext=0:0.25:0.75 % the value of external resistance is varied Tstart=-(3/ws)*((r2d+r_ext)/1)*v1^2/((r1+(r2d+r_ext)/1)^2 + (x1+x2d)^2) smax=(r2d+r_ext)/sqrt(r1^2+(x1+x2d)^2) for j = 1:length(w) s(j)=(ws-w(j))/ws; Tem(j)=-(3/ws)*((r2d+r_ext)/s(j))*v1^2/((r1+(r2d+r_ext)/s(j)) ^2+(x1+x2d)^2); end plot(w,Tem,'k','linewidth',2) hold on; end grid on; xlabel('Speed') ylabel('Electromagnetic Torque') axis([0 2*ws 1.1*Tmax_gen 1.1*Tmax]) gtext('R_e_x_t=0') gtext('R_e_x_t^,=0.25') gtext('R_e_x_t^,^,=0.5') gtext('R_e_x_t^,^,^,=0.75')

The results are tabulated in Table 6.2. Table 6.2 presents the results of Example 6.1. Figure 6.23 depicts the schematic presentation of torque as a function of various external resistances with generator convention. The operation of an induction generator is the same as an induction motor except that the direction of power flow is from wind power driving the shaft of the machine. Therefore, an induction generator injects or supplies power to the local power grid. For motor convention, the positive current flows from the source to the motor. For generator convention, the positive current flows from the motor terminal voltage to the local power grid. This means the sign of the current with motor convention will be negative for generator operation. TABLE 6.2 The Results of Example 6.1 External Resistance (Ω) 0.00 0.25 0.50 0.75

Starting Torque (N-m)

Slip at Maximum Torque (N-m)

Maximum Torque (N-m)

−2.62 −7.06 −10.43 −12.70

0.08 0.24 0.39 0.55

−14.84 (motor) 19.04 (generator)

THE OPERATION OF AN INDUCTION GENERATOR

20

Rext = 0 R′ext = 0.25 R″ext = 0.25

359

R‴ext = 0.75

15

Electromagnetic torque

10

5

0

–5

–10

–15 0 1

200 0.778

400 0.556

600 0.333

800 0.111

1000 –0.111

1200 –0.333

1400 –0.556

1600 –0.778

1800 Speed slip –1

Figure 6.23 The schematic presentation of torque as a function of various external resistances with generator convention.

Example 6.2 For the machine of Example 6.1 with the same supply voltage connected to the local utility, plot the torque–slip characteristics in the speed range of 1000–2000 rpm with motor convention. Write a MATLAB M-file testbed and plot. Figure 6.24 the plot of torque versus speed of the induction machine of Example 6.2. Solution The MATLAB M-file testbed for Example 6.2 is given below. %TORQUE vs SPEED clc; clear all; v1=220/sqrt(3); f=60; P=8; r1=0.2; x1=0.8; r2d=0.13; x2d=0.8; % The electrical quantities are defined ws=120*f/P; w=-1000:0.2:2000; for j = 1:length(w) s(j)=(ws-w(j))/ws; Tem(j)=(3/ws)*(r2d/s(j))*v1^2/((r1+r2d/s(j))^2+(x1+x2d)^2); end

360

MICROGRID WIND ENERGY SYSTEMS

15

10

Electromagnetic torque

5

0

–5

–10

– 15

– 20 –1000 2.111

Figure 6.24

–500 1.556

0 1

500 0.444

1000 – 0.111

1500 – 0.667

2000 Speed – 1.222 slip

The plot of torque versus speed of the induction machine of Example 6.2.

plot(w,Tem,'k','linewidth',2) hold on; grid on; xlabel('Speed') ylabel('Electromagnetic Torque')

The following observations can be made from the above examples. When ωsyn and ωm are rotating in the same direction and ωsyn is rotating faster than ωm, this condition describes the normal operation of the induction machine as a motor. In this region, the slip as given by Equation (6.70) is positive because both are rotating in the same direction: s= Reff =

ωsyn − ωm ωsyn

(6.74)

1 −s R s 2

(6.75)

When ωsyn and ωm are rotating in the same direction and ωsyn is rotating slower than ωm, this operating condition describes the operation of a machine in generator mode. In this region, ωsyn and ωm are rotating in the same direction. However ωsyn is rotating slower than ωm. Therefore, in this region, the mechanical power supplied to the shaft by an external source and ωm is greater than ωsyn and slip is negative (s < 0). This region (s < 0) corresponds to the

THE OPERATION OF AN INDUCTION GENERATOR

361

generator operation. For this region, the equivalent effective resistance as given by Equation (6.71) of the rotor is negative, and the corresponding power (torque) is also negative. This means that the mechanical power is driving the machine and the machine, in turn, delivers electric power at its stator terminals to the source. The generator operation can be summarized as follows: (i) ωm > ωsyn. (ii) ωm and ωsyn are rotating in the same direction. (iii) Generator action: Electrical power is supplied to the network via stator terminals. ωsyn − ωm (iv) s = 1 and power loss is negative, indicating that mechanical energy is converted into electric energy. The power fed from the stator and the power fed from the rotor are both lost as heat in the rotor resistance. This region is called the braking region.

ωsyn N Tm ωm

S

N

Text

Tm ωm

S Text ωsyn

Figure 6.25 The operation of an induction machine as a generator.

362

MICROGRID WIND ENERGY SYSTEMS

Example 6.3 The air gap power of an eight-pole 60 Hz induction machine running at 1000 rpm is 3 kW. What are the rotor copper losses? Solution Figure 6.26 depicts a single-phase equivalent circuit of an induction generator. The air gap power is P Input power to rotor = PAGϕ = 3 I2

2

R2 s

The rotor copper loss is given by Protor loss = 3 I2 2 R2 Therefore, Protor loss 3 I2 2 R 2 =s = Protor in = PAG 2 R2 3 I2 s The rotor power loss, Protor loss, is slip times the air gap power: Protor loss = sPAG For motor convention, P > 0 indicates the machine is consuming power. For generator convention, P > 0 indicates the machine is generating power. The synchronous speed is given by Nsyn = 120 = 120

P

+

IG

Q

R1

f P 60 = 900 rpm 8

R′2

jX1

+

Im PAG

Vgrid –

jX′2

Vind

R′2

(1 – s) s

– Tem

TL

Figure 6.26 A single-phase equivalent circuit of an induction generator.

THE OPERATION OF AN INDUCTION GENERATOR

363

The slip is given by s= =

Nsyn −Nm Nsyn 900−1000 = − 0 11 900

The machine is operating as an induction generator. Therefore, following generator convention, reversing the direction of air gap power, the rotor power loss is given by Protor loss = −sPAG = 0 111 × 3000 = 333 kW Example 6.4 A three-phase, six-pole, Y-connected induction generator rated at 400 V, 60 Hz, is running at 1500 rpm supplying a current of 60 A at a power factor of 0.866 leading. It is operating in parallel with a local power grid. The stator copper losses are 2700 W; rotational losses are 3600 W. Perform the following: (i) Determine the active and reactive power flow between the induction generator and the power grid. (ii) Determine how much reactive power must be supplied at the induction generator to operate the induction generator at unity power factor. (iii) Calculate the efficiency of the generator. (iv) Find the value of stator resistance, rotor resistance, and the sum of the stator and rotor reactance. (v) Compute the electromechanical power developed by the rotor. Solution The synchronous speed is given by Nsyn = 120 = 120

f P 60 = 1200 rpm 6

The slip is given by s=

Nsyn −Nm 1200 − 1500 = −0 25 = 1200 Nsyn

364

MICROGRID WIND ENERGY SYSTEMS

The active power at the stator terminals is given by Pgrid = 3VL − N Is cos θ =3×

400 3

× 60 × 0 866 = 36, 000 W

The reactive power at the stator terminals is given by Qgrid = 3VL − N Is sin cos − 1 θ =3×

400 3

× 60 × sin cos − 1 0 866 = 20,786 VAr

For unity power factor operation, the reactive power that needs to be supplied locally is the same as Qgrid = 20,786 VAr The stator copper loss is PR1 = 2700 W The air gap power is PAG = Pgrid + PR1 = 36, 000 + 2700 = 38,700 W 3ϕ The fixed loss is Protational loss = 3600 W To compute the mechanical inpurt power, let PG be the electromechanical power developed in the rotor: Pmech = PG + Protational = 1 −s PAG + Protational = 1 − − 0 25 38, 700 + 3600 = 51,975 W The efficiency is η=

Pelec 36,000 = 0 6276 = 69 26 = Pmech 51,975

THE OPERATION OF AN INDUCTION GENERATOR

365

The stator resistance is given by PR1 3I 2

R1 =

2700 = 0 25 Ω 3 × 602

=

Following generator convention, the rotor copper loss is given by PR2 = − sPAG = − − 0 25 × 38,700 = 9675 W The rotor resistance is given by PR2 3I 2

R2 =

9675 = 0 90 Ω 3 × 602

=

The sum of the reactance of the rotor and the stator is given by X = X 1 + X2 = =

Qgrid 3I 2

20, 786 = 1 92 Ω 3 × 602

The electromechanical power developed by the rotor is = − 3I 2

1−s R s 2

= − 3 × 602 ×

1 − − 0 25 × 0 9 = 48,600 W − 0 25

Figure 6.27 depicts a power flow diagram for the generator mode of the operation of an induction machine.

Pgrid

PR1 = I12 . R1

PAG = PG – PR2

PR2 = I22 . R′2

PG = PAG (1 – s)

Pmech

Rotational losses

Figure 6.27 A power flow diagram for the generator mode of the operation of an induction machine.

366

6.7

MICROGRID WIND ENERGY SYSTEMS

DYNAMIC PERFORMANCE

In the previous sections, we analyzed the steady state of induction machines. For dynamic analysis, we must model the machines by a set of differential equations. For stator windings, we have three coupled windings that are sinusoidally distributed around the stator. When these coupled windings are represented based on self-inductance and mutual inductances, they give rise to a set of three time-varying differential equations. Similarly, we can obtain three time-varying differential equations for the rotor windings. The electromagnetic torque can be expressed by a nonlinear algebraic equation, and the rotor s dynamic can be represented by a differential equation expressing the rotational speed of the motor. Therefore, the induction machine dynamic performance can be expressed by seven differential equations and one algebraic equation. The dynamic modeling of an induction machine is an advanced concept that students need to study with additional coursework.14,15 Here it is instructive to study the results of a dynamic analysis as depicted in Figures 6.28 and 6.29. Figure 6.28 depicts the stator current start-up condition of an induction machine. As expected, the machine stator current has many cycles of transient oscillation before it reaches the steady-state current. The steady-state current supplied by the source magnetizes the machine and is lost as heat because the machine is operating at no load. Figure 6.29 depicts the machine shaft speed from rest (start-up) to no-load speed that is just below the synchronous speed. Figure 6.30 depicts the transient oscillations of a machine. In Figure 6.30, the machine goes through 0.4 seconds of oscillations and reaches its maximum torque. The normal region of the induction machine operation is below

Stator phase (a) current

120 100 80 60 40 20 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (second)

Figure 6.28 The induction machine stator current for a no-load start-up.

5

367

DYNAMIC PERFORMANCE

400 350 Shaft speed

300 250 200 150 100 50 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (second)

Figure 6.29 Induction machine shaft speed for no-load start-up.

Induction motor in balanced conditions (stator reference frame) 4

T st rans ar ie t-u n p to

sc

Tmax

illa

tio

3

n

o al rm No nge ra

du

rin

g

Te (p.u.)

g

tin

ra

pe

2

1 0

–1

0

0.2

0.4

0.6 0.8 Time (second)

1

1.2

1.4

Figure 6.30 The dynamic performance of an induction machine.

maximum speed. In Figure 6.30, the motor operation of the machine is presented. Because the machine is simulated at no load from rest, the machine develops enough torque to support the machine resistive and rotational losses. The schematic of a microgrid of an induction machine controlled as a generator is presented in Figure 6.31.

368

MICROGRID WIND ENERGY SYSTEMS

Local power grid

Smart metering

Stator winding

GB

Rotor winding

Variable resistor

Firing unit

Control circuit

Optical coupling

Rotor

Figure 6.31 A microgrid of an induction machine controlled as an induction generator supplying power to the local power grid.

Example 6.5 ing data:

Consider the microgrid given in Figure 6.32 with the follow-

I. Transformer rated at 440 V/11 kV, with the reactance of 0.16 Ω and resistance of 0.02 Ω, and rated at 60 kVA. II. A three-phase, eight-pole induction machine rated at 440 V, 60 Hz and 50 kVA, 440 V, 60 Hz, with stator resistance of 0.2 Ω/phase, rotor referred resistance in the stator side of 0.2 Ω/phase, stator reactance of 1.6 Ω/phase, and rotor referred reactance of 0.8 Ω/phase. The generator speed is 1200 rpm. Perform the following: (i) Give the per unit (p.u.) model of the system. (ii) Compute the power delivered to the local power grid. (iii) Compute the reactive power flow between the grid and the induction generator. Assume base values equal to the rating of the induction machine.

11 kV (infinite bus)

Transformer LV TL

Ind gen

Tm

Figure 6.32 The microgrid connected to a local power grid for Example 6.5.

DYNAMIC PERFORMANCE

Solution I. By selecting the induction machine’s rating as a base, Sb = 50 kVA Vb = 440 V The base impedance of the system is

Zb =

Vb2 4402 = 3 872 = Sb 50 × 103

The p.u. value of transformer resistance is Rtran 0 02 = = 0 005 3 872 Zb

Rtran, p u =

The p.u. value of transformer reactance is

Xtran, p u =

Xtran 0 16 = 0 04 = 3 872 Zb

The p.u. value of stator resistance is Rs , p u =

Rs 02 = = 0 052 Zb 3 872

The p.u. value of rotor resistance referred to primary is

Rr, p u =

Rr 02 = 0 052 = Zb 3 872

The p.u. value of stator reactance is Xs, p u =

Xs 16 = = 0 413 Zb 3 872

The p.u. value of rotor resistance referred to primary is Xr , p u =

Xr 08 = = 0 207 Zb 3 872

369

370

MICROGRID WIND ENERGY SYSTEMS

II. The synchronous speed, Ns, is Ns =

120f 120 × 60 = = 900 rpm P 8

The slip, s, is s=

Ns −N 900− 1200 = = −0 333 Ns 900

where N is the motor shaft (rotor) speed in rpm. The rotor voltage frequency is fr = s f s fr = 0 333 × 60 = 20 Hz The supply voltage is 440 V = 1 p.u. The base current is Ib =

VAb 3Vb

=

50 × 103 3 × 440

= 65 61 A

p.u. impedance Zp u = =

Rs, p u + Rtran, p u + Rr, p u s

2

+ Xs, p u + Xtran, p u + Xr, p u

2

0 052 + 0 005− 0 052 0 333 2 + 0 413 + 0 04 + 0 207 2 = 0 667

The power factor angle tan − 1

Xs, p u + Xtran, p u + Xr, p u 0 413 + 0 04 + 0 207 = tan − 1 = 98 54 Rs, p u + Rtran, p u + Rr, p u s 0 052 + 0 005−0 052 0 333

Zp u = Zp u ∠ θ = 0 667 ∠ 98 54 The p.u. stator current in the motor convention is Ip u =

Vb 1 = = 1 499 ∠ − 98 54 Zp u 0 667 ∠ 98 54

The actual value of current in the motor convention is I m = Ib × I p u = 65 61 × 1 499 = 98 35 ∠ −98 54 A Therefore, Im = 98.35 ∠ −98.54 A.

DYNAMIC PERFORMANCE

371

Because this angle is more than 90 , the power flow is from the induction generator to the local power grid. For generator convention, the direction of current is reversed to represent IG as in Figure 6.26. Therefore, the current in generator convention is IG = 98 35 ∠ 180 −98 54 = 98 35 ∠ 81 46 A The active power input to the grid is Pgrid = 3 Vgrid IG cos θ = 3 × 440 × 98 35 × cos81 46 = 11,130 W The active power lost in the transformer is 2 Ploss = 3IG Rtrans = 3 × 98 352 × 0 02 = 580 W

Therefore, the active power injected by the induction generator to the transformer is P = Pgrid + Ploss = 11,130 + 580 = 11,710 W The reactive power input to the grid is Qgrid = 3 Vgrid IG sin θ = 3 × 440 × 98 35 × sin 81 46 = 74,121 W The reactive power lost in the transformer is 2 Xtrans = 3 × 98 352 × 0 15 = 4355 W Qloss = 3IG

Therefore, the reactive power consumed by the induction generator is given by Q = Qgrid −Qloss = 74,121 −4355 = 69,766 W Example 6.6 Consider a three-phase Y-wound rotor-connected induction generator rated 220 V, 60 Hz, and 16 hp; the poles of the machine can be changed from 2 to 12 to control the wind speed. The machine has the following parameters: R1 = 0.2 Ω/phase and X1 = 0.4 Ω/phase. R2 = 0.13 Ω/phase and X2 = 0.4 Ω/phase. Plot the power versus speed curve of the machine controlling the wind speed by changing the number of poles.

372

MICROGRID WIND ENERGY SYSTEMS

Solution The following MATLAB M-file testbed depicts the operation of the machine. %POWER vs SPEED clc; v1=220/sqrt(3); f=60; P=2; r1=0.2; x1=0.4; r2d=0.13; x2d=0.4;

% The electrical quantities are defined

for P=2:2:12 % The no. of poles is varied from 2 to 12 ws=120*f/P; w=0:.2:7200; % The value of speed is varied till synchronous speed for i=1:length(w) s(i)=(ws-w(i))/ws; Tem(i)=(3/ws)*(r2d/s(i))*v1^2/((r1+r2d/s(i))^2+(x1+x2d)^2); Po(i)=-Tem(i)*w(i)/1000; % Power in kW end plot(w,Po) hold on; end axis([0 7200 0 50]) grid on; set(gca,'XDir','reverse') xlabel('Speed (rpm)') ylabel('Power (kW)')

Figure 6.33 depicts the power versus speed of a variable pole induction generator for various rotor speeds. 6.8

THE DOUBLY FED INDUCTION GENERATOR

Electric machines are classified based on the number of windings in the conversion of mechanical power to electric power. A singly fed machine has one winding. The SCIG-type machines have one winding, which contributes to the energy conversion process. Doubly fed machines have two windings that likewise are instrumental in energy conversion. The wound rotor doubly fed induction generator (DFIG) is the only electric machine that can operate with rated torque to twice the synchronous speed for a given frequency of operation. In a DFIG, the current flowing in the magnetizing branch and torque current are orthogonal vectors. It is not desirable to design machines that are magnetized from the rotor because commutation systems, supporting slip rings, and brushes are needed to inject current into the rotor

THE DOUBLY FED INDUCTION GENERATOR

50 Nr >1800

Nr >3600

45

373

Nr Nr Nr Nr >1200 >900 >720 >600

40

Power (kw)

35 30 25 20 15 10 5 0

7000

6000

5000

4000

3000

2000

1000

0

Speed (rpm)

Figure 6.33 Power versus speed of a variable pole induction generator for various rotor speeds.

winding. These types of machines have high maintenance costs. However, in these machines, the stators can have a unity power factor. The frequency and the magnitude of the rotor voltage are proportional to the slip as expressed by Equation (6.30). In principle, the DFIG is a transformer at a standstill. If the DFIG is producing torque and operating as a motor, the rotor is consuming power. At a standstill, all power fed into the stator is consumed as heat in the stator and the rotor. Therefore, at low speeds, the efficiency of the DFIG is very low because the supplied current is mainly used to produce magnetizing current and the power conversion processes as a function of a motor or a generator do not take place. If the DFIG is operating at above the synchronous speed, the mechanical power is fed in both through the stator and rotor. Therefore, the machine has higher efficiency, and the machine can produce twice the power as a singly fed electric machine. With the DFIG at below synchronous speeds, the stator winding is producing electric power, and part of its power is fed back to the rotor. At speeds above synchronous speeds, the rotor winding and stator winding are supplying electric power to the grid. However, DFIGs do not produce higher torque per volume than singly fed machines. The higher power rating can be obtained because of the higher speed and without weakening the magnetic flux. A DFIG configuration system is depicted in Figure 6.34. This DFIG is a wound rotor induction generator (WRIG) with the stator windings directly connected to the power grid. The DFIG has two parallel AC/DC converter units. Although these converters act together, they are not necessarily fully identical about their power rating.

374

MICROGRID WIND ENERGY SYSTEMS Pstator > 0

Pmec

Pgrid ≈ Pstator + Protor WRIG Grid

Grid side converter

Rotor side converter AC

DC AC

DC

Sub-synchronous

Over-synchronous

Protor < 0

Protor > 0

Figure 6.34 A microgrid of a doubly fed wound induction generator.

The rotor windings connected to an AC/DC power convert on the grid side and a DC/AC power converter on the rotor side. The back-to-back converter operates as a bidirectional power converter with a common DC bus. The transformer in Figure 6.35 has two secondary windings: one winding is connecting the stator and the other connecting the rotor. The converter on the rotor side makes it possible to operate the rotor excitation at a lower DC bus voltage. This DFIG provides reactive power control through its power converter because it decouples active and reactive power control by

Net metering

Infinite bus

Infinite bus

Net metering Local load

Local load Grid-side converter

Control-side converter

AC/DC

Power winding

Rotor

DC/AC Control winding

V,i

Control system

i ωm

Speed sensor

V,i

Figure 6.35

A microgrid of brushless doubly fed induction generator.

BRUSHLESS DOUBLY FED INDUCTION GENERATOR SYSTEMS

375

independently controlling the rotor excitation current. A DFIG can supply (absorb) reactive power to and from the power grid. The control method is based on the variable speed/variable pitch wind. Two hierarchical control levels are used. These controllers are designed to track the wind turbine operation point to limit the turbine operation in the case of high wind speeds. Also, the controller controls the reactive power injected into the power grid and the reactive power consumed by the WTG. The power supervisory controller controls the pitch angle to keep the wind turbine operating at the rated power. At the same time, the speed controller controls the shaft speed of the generator to ensure that it remains within a safe range. However, at low wind speeds, the speed controller attempts to maximize the generated power and generator efficiency. Therefore, the change in generator speed follows the slow change in wind speed. As an inherent part of DFIG systems, the generator stator feeds up to 70% of the generated power directly into the grid, usually at low voltage, using a step-up transformer. A well-established disadvantage of DFIG systems is the occurrence of internal stray currents in the generator. These currents accelerate generator bearing failure. Protective counter methods include the design of special generator bearings and seals that shield the bearings against the negative impact of stray currents. A wound rotor DFIG has several advantages over a conventional induction generator. Because a power converter actively controls the rotor winding, the induction generator can generate and consume reactive power. Therefore, the DFIG can support the power system stability during severe voltage disturbances by providing reactive power support. As an inherent part of DFIG systems, the generator stator feeds the remaining 70–76% of total power directly into the grid, usually for up to 690 V. The wind microgrid is connected to the local grid using a step-up transformer. The DFIG systems are designed for both 60 and 50 Hz systems, but each grid situation requires a generator operation adapted to the specific operational circumstances. However, the control of DC/AC converters is an advanced topic that is not the subject of this book. Students who are interested in the control of converters in green energy systems should refer to Reference 17. SCIG have a lower cost of manufacturing than WRIG and are relatively simple to design; they are robust, cost effective, and widely used in wind microgrids.

6.9

BRUSHLESS DOUBLY FED INDUCTION GENERATOR SYSTEMS

Brushless DFIG systems18 are designed by placing two multiphase winding sets with a different number of pole pairs on the stator structure. One of the stator winding sets is designated as power winding and is connected to the power grid. The second winding control is supplied from a converter and controls the energy conversion process. The generator is controlled by varying the frequency of the winding connected to the power converter. Because the pole

376

MICROGRID WIND ENERGY SYSTEMS

pairs of two windings are not identical, the low-frequency magnetic induction is created in the winding that connects to the power grid over a speed range that the rotor supplies by wind power. Figure 6.35 depicts a microgrid of brushless DFIG. Brushless DFIGs do not utilize the magnetic core efficiently. The dualwinding set stator area is physically larger than that of other electric machines of comparable power rating.

6.10 VARIABLE-SPEED PERMANENT MAGNET GENERATORS These types of wind generators operate at variable wind speed.19 They use a “full” AC/DC and DC/AC power converter. The DC power is inverted using a DC/AC inverter and is connected to a step-up transformer and then connected to the local power grid as shown in Figure 6.36. The variable-speed permanent magnet generator of Figure 6.37 produces a variable AC voltage with a variable frequency. Because the power generated is not at the frequency of the local power grid, the output power of the variable frequency generator cannot be injected into local power grid as we have discussed in Section 4.6. Therefore, the variable AC frequency voltage is rectified by the AC/DC rectifier (see Figure 6.37). The DC bus of Figure 6.36 can be used to charge a storage system using a boost–buck converter. Figure 6.38 depicts a variable-speed generator. This type of generator is studied in Chapter 4. The field winding of the generator of Figure 6.37 is supplied from DC power using an AC/DC rectifier from an AC bus of the microgrid. Because the supplied wind mechanical power has a variable speed, the generator output power would also contain variable frequency. The DC/AC inverter is used to convert the DC power to AC power at the local power grid frequency and voltage as shown in Figures 6.36 and 6.37. The microgrid of wind power can be connected to the local power grid of an AC bus because it is operating at the frequency of the local power grid. However, the coordinated control of converters of Figures 6.36 and 6.37 is an advanced topic that is not the subject

N AC/DC

DC/AC Buck / boost

S Battery storage

Net metering

Local power grid

Local loads

DC/DC DC bus

Figure 6.36 A microgrid of a variable-speed permanent magnet for a wind generator.

A VARIABLE-SPEED SYNCHRONOUS GENERATOR

377

Rectifier DC AC DC bus Inverter

Rectifier AC

Pitch

Variable speed synchronous generator

Local power grid

DC DC

AC

Boost/buck converter

P

Q

DC

Battery storage

DC

Flywheel

Figure 6.37 A microgrid of a multipole synchronous generator.

Rectifier DC AC

Rectifier Local power grid

Pitch

Variable speed synchronous generator

Figure 6.38 A variable-speed synchronous generator.

of this book. Students who are interested in the control of converters in green energy systems should refer to References 17 and 20.

6.11 A VARIABLE-SPEED SYNCHRONOUS GENERATOR The rotor of a synchronous generator rotates at synchronous speed.16,18 For a synchronous generator, the frequency of the voltage induced in the stator windings is given by the expression below: ωsyn =

2 ωs P

378

MICROGRID WIND ENERGY SYSTEMS

Local utility Net metering

Infinite bus

Local loads

Exciter machine

DFIG

AC/DC convt. no.1

DC/AC convt. no.2

Figure 6.39 A microgrid of variable-speed wind turbine generator with the converter isolated from the grid.

If P = 2, then ωsyn = ωs. And the frequency of the induced voltage is given by ωs = 2 π fe If the wind speed is variable, the induced voltage will also be time varying, and it will be of multiple frequencies. Figure 6.39 depicts such a wind-based microgrid. However, before connecting the generator to the local power grid, we must operate the generator at the synchronous speed. Depending on the expected wind speed for a given location, the generator can be designed with a gear system. The gear ratio is adjusted such that the speed of the rotor of the generator is at synchronous speed. Therefore, the induced voltage of the stator of the generator is at the same frequency as the local power grid.

6.12 A VARIABLE-SPEED GENERATOR WITH A CONVERTER ISOLATED FROM THE GRID Another type of WTG system consists of an electrical exciter machine together with a DFIG. In comparison with a normal DFIG system, this wind generator has one converter. By including the excitation machine, it is possible to isolate the power converter, and it is not directly connected to the grid. That is, the stator is the only grid-connected output. This solution is different from a normal DFIG grid connection. The generator rotor power is fed into the grid via a power converter.

A VARIABLE-SPEED GENERATOR WITH A CONVERTER ISOLATED FROM THE GRID

379

Figure 6.39 depicts a variable-speed WTG with a converter isolated from the grid. The first converter is a DC/AC inverter that feeds the rotor of DGIF. However, in this topology, the second converter unit is an AC/DC rectifier, which is supplied from the exciter machine. Furthermore, in contrast to both synchronous and asynchronous generators, DFIGs are inherently incapable of acting as electric brakes for the sudden separation of a wind turbine from the grid or sudden high wind speed. However, in the above topology, the exciter machine power can be used to drive an electric brake. The electric brake may also be used together with aerodynamic braking, minimizing peak torque loads. These wind turbines are characterized by lower inertia than classical power plants; therefore, they cannot participate in power system load frequency regulation. When the wind generators are equipped with a storage system, they can participate in load frequency control. The variable-speed turbines are designed based on the use of back-to-back power electronic converters. The intermediate DC voltage bus creates an electrical decoupling between the machine and the grid. Such decoupling creates a new opportunity to use these types of wind generating systems for load frequency control. Example 6.7 Select AC/DC rectifiers and DC/AC inverters for a 600 kW variable-speed wind generator operating at 690 V AC. The utility-side voltage is 1000 V. Solution The peak value of the instantaneous line-to-line as voltage is VL− L, peak = 2 VL −L, rms = 2 × 690 = 975 8 V Therefore, the DC-side voltage rating of the rectifier is ≥976.8 V. Let the voltage rating of the rectifier be 1000 V on the DC side and 690 V on the AC side. The inverter also can be selected with a rating of 1000 V on the AC side. Both the rectifier and the inverter should be rated at 600 kW. In this chapter, we have studied the modeling of induction machines and their operation as motors and generators. The use of induction generators as a source of power in microgrids requires an excitation current for generator operation to be provided from local microgrids. If the wind-based microgrids are connected to the local power grids, the power grids will provide the excitation currents (VAr). Therefore, the local power grids must be planned to provide the reactive power (VAr) requirements of the wind microgrids. We have also reviewed DFIGs, variable-speed induction generators, and variable-speed permanent magnet generators.17–19 The coordinated control of the converter is an advanced topic that is not the subject of this book. Students who are interested in the control of converters in green energy systems should refer to Reference 17.

380

MICROGRID WIND ENERGY SYSTEMS

PROBLEMS 6.1

Consider a wind microgrid given in Figure 6.40. The system has a local load rated at 100 kVA at a power factor rated 0.8 lagging. The three-phase transformer is rated 11 kV/0.44 kV; 300 kVA; X = 0.06 p.u. A three-phase, eight-pole induction generator is rated at 440 V, 60 Hz, with stator resistance of 0.08 Ω/phase, rotor referred resistance in the stator side of 0.07 Ω/phase, stator reactance of 0.2 Ω/phase, and rotor referred reactance of X2 0.1 Ω/phase. Compute the following: (i) The per unit equivalent model based on a kVA base of 300 kVA and 440 V. (ii) The shaft mechanical power if the shaft speed is at 1200 rpm. (iii) The amount of power injected into the local utility. (iv) The flow of reactive power between grid and local microgrid. (v) Compute the amount of the reactive power that must be placed at the local grid to have unity power factor at the local power grid.

6.2

The microgrid of Figure 6.41 is supplied by an induction generator. The system has a local load rated 100 kVA at a power factor rated 0.8 lagging. The three-phase transformer is rated at 11/0.44 kV, 300 kVA, and reactance of 6%. A three-phase, eight-pole induction machine rated at 440 V, 60 Hz and 500 kVA, 440 V, 60 Hz, with stator resistance of 0.1 Ω/phase, rotor referred resistance in the stator side of 0.1 Ω/phase, stator reactance of 0.8 Ω/phase, and rotor referred reactance of 0.4 Ω/ phase. Compute the following: (i) The per unit power flow model and short-circuit model based on a base of 500 kVA and 440 V. (ii) If the speed of the induction generator is 1000 rpm, what is the rotor frequency? 11 kV (infinite bus)

Transformer

Load LV

Gen TL Tm

Figure 6.40 Schematic of Problem 6.1.

PROBLEMS

381

Local power grid 11 kV

Load

Transformer

LV

TL

Gen

Tm

Figure 6.41 System of Problem 6.2.

(iii) The flow of active and reactive power between the microgrid and the local power grid. 6.3

A six-pole WRIG is rated at 60 Hz, 380 V, 160 kVA. The induction machine had a stator and referred rotor resistance of 0.8 Ω/phase and stator and rotor reactance of 0.6 Ω/phase. The generator shaft speed is at 1500 rpm. Determine how much resistance must be inserted in the rotor circuit to operate the generator at 1800 rpm.

6.4

A 400 V three-phase Y-connected induction generator has the following data: Z1 = 0 6 + j1 2 Ω phase Z2 = 0 5 + j1 3 Ω phase The generator is connected to a local power grid. Perform the following: (i) The maximum active power that generator can supply. (ii) The reactive power flow between the induction generator and the local power grid.

6.5

Design a 15 kW wind power generator that is supplied from variable wind speed. The designed system must provide 220 VAC, single-phase AC power. Compute the DC bus voltage.

6.6

The same as problem 6.5, except the wind generating system must provide three-phase AC nominal voltage of 210 VAC. Compute the DC bus voltage.

6.7

Design 2 MW wind systems using a variable-speed system. The DC bus voltage is to be a nominal value of 600 VDC. The generators are located

382

MICROGRID WIND ENERGY SYSTEMS

TABLE 6.3 13.2–132 kV Class One Phase-Neutral Return Line Model

Conductor Magpie Squirrel Gopher

DC Resistance (Ω/km)

Reactance (Ω/km), XL

Susceptance (S/km), YC

Current Ratings (A)

1.646 1.3677 1.0933

j0.755 j0.78 j0.711

j1.45e−7 j6.9e−7 j7.7e−7

100 130 150

at 5 miles from the local utility. The utility voltage is three-phase AC rated at 34.5 kV. The data for the transmission line is given in Table 6.3. The data for transformers are given as 460 V/13.2 kV, 250 kVA 10% impedance and 13.2–34.5 kV, 1 MVA 8.5% impedance. Perform the following: (i) Give the one-line diagram. (ii) Per unit model base on rated wing generator. 6.8

A wound rotor six-pole 60 Hz induction generator has a stator resistance of 1.1 Ω/phase and rotor resistance of 0.8 Ω and runs at 1350 rpm. The prime mover torque remains constant at all speeds. How much resistance must be inserted in the rotor circuit to change the speed to 1800 rpm? Neglect the motor leakage reactances, X1 and X2.

6.9

Consider a three-phase Y-wound rotor-connected induction generator rated 220 V, 60 Hz, 16 hp, with eight poles with the following parameters: R1 = 1 Ω/phase and X1 = 1.6 Ω/phase. R2 = 0.36 Ω/phase and X2 = 1.8 Ω/phase. The generator is connected to the local power grid. Write a MATLAB simulation testbed to plot slip and speed as a function of machine torque and various external inserted resistance in the rotor circuit. Make the plot for a value of external resistance of 0.0, 0.4, 0.8, and 1.2 Ω.

6.10

The Rayleigh distribution functions for different mean wind speeds are shown in Figure 6.42 (see Appendix D). The curve moves to the right for greater mean wind speeds (also greater values of the shape parameter (a), which means more days have high winds—hence potentially more wind energy revenue). The wind speed frequency distributions shown in Figure 6.43 are obtained from wind speed data for a year of 10 minute means for two sites with similar average wind speed. (i) Compute the mechanical power generated for an hour of operation assuming v = 12 m s and the air density is 1.2 kg/m3.

3 m/s

Relative frequency

0.25 0.2 0.15

6 m/s

0.1

9 m/s

0.05 0

0

1

2

3

4

5 6 7 8 9 10 11 12 13 14 15 Wind speed (m / s)

18

18

12

12 Percent

Percent

Figure 6.42 Rayleigh distribution functions for three different mean wind speeds.

6

6

0

1

0 3 5 7 9 11 13 15 17 19 21 23 24 Wind speed (m/s)

1 3 5 7 9 11 13 15 17 19 21 23 25 Wind speed (m/s)

Figure 6.43 Rayleigh wind speed frequency distribution functions for two sites each having the same average wind speeds. 0.06

Relative frequency

0.05

0.04

0.03

0.02

0.01

0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Wind speed (m/s)

Figure 6.44 Wind speed data for Problem 6.11.

384

6.11

MICROGRID WIND ENERGY SYSTEMS

Assuming the annual wind speed data for a site are given in Figure 6.44 with an annual mean speed of v = 13 29 m s, it is required to estimate the average annual power and energy density for that site.

REFERENCES 1. Wikipedia. History of wind power. Available at http://en.wikipedia.org/wiki/ Wind_Energy#History. Accessed August 10, 2009. 2. The Technical University of Denmark, National Laboratory for Sustainable Energy. Wind Energy. Available at http://www.risoe.dk/Research/sustainable_energy/wind_energy.aspx. Accessed July 11, 2009. 3. Justus, C.G., Hargraves, W.R., Mikhail, A., and Graber, D. (1978) Methods of estimating wind speed frequency distribution. Journal of Applied Meteorology, 17(3), 350–353. 4. California Energy Commission, Energy Quest. Chapter 6: wind energy. Available at http://www.energyquest.ca.gov/story/chapter16.html. Accessed June 10, 2009. 5. Jangamshetti, S.H. and Guruprasada Rau, V. (1999) Site matching wind turbine generators: a case study. IEEE Transactions on Energy Conversion, 14(4), 1537–1543. 6. Jangamshetti, S.H. and Guruprasada Rau, V. (2001) Optimum siting of wind turbine generators. IEEE Transactions on Energy Conversion, 16(1), 8–13. 7. Quaschning, V. (2006) Understanding Renewable Energy Systems, Earthscan, London. 8. Freris, L. and Infield, D. (2008) Renewable Energy in Power Systems, Wiley, Hoboken, NJ. 9. Patel, M.K. (2006) Wind and Solar Power Systems: Design, Analysis, and Operation, CRC Press, Boca Raton, FL. 10. Hau, E. (2006) Wind Turbines: Fundamentals, Technologies, Application and Economics, Springer, Heidelberg. 11. Simoes, M.G. and Farrat, F.A. (2008) Alternative Energy Systems: Design and Analysis with Induction Generators, CRC Press, Boca Raton, FL. 12. AC motor theory. Available at http://www.pdftop.com/ebook/ac+motor+theory/. Accessed December 5, 2010. 13. U.S. Department of Energy, National Renewable Energy Laboratory. Available at http://www.nrel.gov/. Accessed October 10, 2010. 14. HSL Automation Ltd. Basic motor theory: squirrel cage induction motor. Available at http://www.hslautomation.com/downloads/tech_notes/HSL_Basic_Motor_Theory. pdf. Accessed October 10, 2010. 15. Krause, P. and Wasynczuk, O. (1989) Electromechanical Motion Devices, McGraw-Hill, New York. 16. Majmudar, H. (1966) Electromechanical Energy Converters, Allyn & Bacon, Reading, MA. 17. Boldea, I. (2005) The Electric Generators Handbook. Variable Speed Generators, CRC Press, Boca Raton, FL.

REFERENCES

385

18. Pena, R., Clare, J.C., and Asher, G.M. (1996) Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed windenergy generation. IEE Proceedings of Electric Power Applications, 143(3), 231–241. 19. 1999 European Wind Energy Conference: Wind energy for the next millennium. Proceedings of the European Wind Energy Conference, Nice, France, 1–5 March 1999 (1999) Earthscan, London. 20. Keyhani, A., Marwali, M., and Dai, M. (2010) Integration of Green and Renewable Energy in Electric Power Systems, Wiley, Hoboken, NJ.

CHAPTER 7

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS 7.1

INTRODUCTION

We have established that the main objectives of electric energy distribution are to provide rated voltage and rated frequency at the bus loads. In Chapter 4, we discussed load frequency control; here we will focus on voltage control. For ensuring the rated voltage at each bus in a power grid, the system is modeled in a steady state. The calculation of the bus load voltage formulated as a power flow problem. Once the load the bus voltage magnitude and phase angle are calculated, then the power flow on lines can be computed from line impedance and voltages on the two ends of a line. The power flow analysis is an engineering tool to ensure proper operation of the power grids. In the formulation of the power flow problem, we will study how interconnected transmission systems are modeled. The bus admittance matrix and bus impedance matrix models are presented in this chapter. At the end of the chapter, we will review three methods for solving power flow problems: Gauss–Seidel, Newton–Raphson, and fast decoupled load flow (FDLF) solutions. Several solved examples and homework problems are provided to further students’ understanding of microgrid design.

Design of Smart Power Grid Renewable Energy Systems, Third Edition. Ali Keyhani. © 2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/smartpowergrid3e

386

VOLTAGE CALCULATION IN POWER GRID ANALYSIS

7.2

387

VOLTAGE CALCULATION IN POWER GRID ANALYSIS

In a circuit problem, the impedance of loads and the source voltage are given. The problem is to find the current flow in the circuit and calculate the voltage across each load. In the voltage calculation of a power flow problem, the loads are given as active and reactive power consumption, and the bus load voltage is to be computed. We can study this problem via two methods: (1) assume the voltages across the loads are specified and calculate the source voltage, and (2) assume the source voltage and compute the bus load voltage (this is known as a power flow or load flow problem). Example 7.1 illustrates the first method. Example 7.1 A three-phase feeder is connected through two cables with an equal impedance of 4 + j15 Ω in series to 2 three-phase loads. The first load is a Y-connected load rated at 440 V, 8 KVA, p.f. = 0.9 (lagging), and the second load is a Δ-connected motor load rated at 440 V, 6 KVA, 0.85 p.f. (lagging). The motor requires a load voltage of 440 V at the end of the line on the Δ-connected loads. Perform the following: (i) Give the one-line diagram. (ii) Find the required feeder voltage.

Solution Figure 7.1 depicts the one-line diagram of a radial feeder. The line voltage at bus 3 of Example 7.1 is equal to 440 V. kVAr3 6000 The rated current drawn by a motor on bus 3 is I3 = = 3 V3 3 × 440 ∠ −cos − 1 0 85 = 7 87 ∠ − 31 77 A. The voltage at bus 2 is given by V2,ph = V3,ph +

z

2−3 × I3

= 440

3+ 4 + j15 ×7 87 ∠ −31 77 = 353 04 ∠ 13 7 V.

The rated current drawn by a load on bus 2 is I2 = ∠ −cos − 1 0 9 = 7 55 ∠ −25 84 A. 1 I1 Local utility

kVAr2 8000 = 3 V2 3 × 353 04

2

3

1–2

2–3

V2 S2

V3

Figure 7.1 The one-line diagram of Example 7.1.

S3

388

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

The supply current of the generator is given by I1 = I2 + I3 = 7.55 ∠ − 25.84 + 7.87 ∠ − 31.77 = 15.39 ∠ − 28.87 A. The generator phase voltage is given by V1 = V2 +

z

1−2 × I1

= 353.04 ∠ 13.7 + (4 + j15) × 15.39 ∠ − 28.87

= 569 21 ∠ 26 73 V The line voltage of the generator is given by V1 3 = 569 21 × 3 = 985 90 V. Example 7.1 is not a practical problem. The generator voltage is controlled by its excitation system. In practice, the field current is set to obtain the rated generator voltage. If the generator has two poles and the generator is operating at synchronous speed, that is, for a 60 Hz system, it operates at 3600 rpm. Therefore, in a practical problem, we need to compute the voltage at load buses given the generator voltage and load power consumption. The solution to this latter problem—known as the power flow problem—is more complex. Example 7.2 Consider a distributed feeder presented in Figure 7.2. Assume the following: (i) Feeder line impedances, that is,

z

1−2

and

z

2−3,

are known.

(ii) The active and reactive power consumed, that is, S2 and S3, by loads are known. (iii) The local power grid bus voltage V1 is known, and all data are in per unit. Solution Let us write Kirchhoff’s current law for each node (bus) of Figure 7.2 and assume that the sum of the currents away from the bus is equal to zero. That is, for buses 1–3, we have v1 − v2 y12 −I1 = 0 (7.1)

v2 − v1 y12 + v2 − v3 y23 + I2 = 0 v3 − v2 y23 + I3 = 0 1

2

3

1–2

2–3

I1 Local utility

V2 S2

Figure 7.2 A distribution feeder.

V3

S3

VOLTAGE CALCULATION IN POWER GRID ANALYSIS

where y12 = 1/

z

1−2

and y23 = 1/

I1 =

S1 V1

z

389

2−3:

∗

S2 V2

, I2 =

∗

, and I3 =

S3 V3

∗

(7.2)

We can rewrite Equation (7.1) as y12 v1 −y12 v2 = I1 −y12 v1 + y12 + y23 v2 − y23 v3 = − I2 −y23 v2 + y23 v3 = − I3 The above can be written as Y11 Y12

0

V1

Y21 Y22 Y23 0

Y23 Y33

I1

V2 =

− I2

V3

− I3

(7.3)

where Y11 = y12, Y12 = − y12, Y21 = − y12, Y22 = y12 + y23, Y23 = − y23, Y32 = − y23, Y33 = y23. The matrix Equation (7.3) represents the bus admittance matrix; it is also the Ybus model for Example 7.2. The Ybus matrix is described as Ibus = Ybus Vbus

(7.4)

If the system has n buses, Ibus is a vector of n × 1 current injection, Vbus is a voltage vector of n × 1, and Ybus is a matrix of n × n. For Example 7.2, we have three buses. Therefore, the Ybus matrix is 3 × 3 as shown in Equation (7.3) or Equation (7.4). Let us continue our discussion for a general case of a power grid with n buses. For each bus, k, we have Sk = Vk Ik∗

k = 1, 2, …, n

(7.5)

And Ik is the current injection into the power grid at bus k. Therefore, from k row of Ybus matrix, we have n

Ik =

Ykj Vj j=1

(7.6)

390

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

By substituting Equation (7.6) in Equation (7.5), we have ∗

n

S k = Vk

Ykj Vj

k = 1, 2, …,n

(7.7)

j=1

For each bus k, we have a complex equation of the form given by Equation (7.7). Therefore, we have n nonlinear complex equations: n

Pk = Re Vk

∗ ∗ Ykj Vj j=1 n

Qk = Im Vk

∗ ∗ Ykj Vj j=1

n

Pk = Vk

Vj Gkj cos θkj + Bkj sinθkj

(7.8)

Vj Gkj sin θkj −Bkj cos θkj

(7.9)

j=1 n

Q k = Vk j=1

where Ykj = Gkj + jBkj, θkj = θk − θj Vj = Vj cos θj + jsin θj Vk = Vk cos θk + jsin θk For Example 7.2, n = 3, we have six nonlinear equations. However, because the power grid bus voltage magnitude is given and used as a reference with a phase angle of zero, we have four nonlinear equations. In Example 7.2, we are given the feeder impedances ( 1−2 and 2−3) and loads (S2 and S3). To find the bus load voltages, we need to solve the four nonlinear equations for V2, V3, θ2, and θ3. After calculating the bus voltages, we can calculate the complex power (S1 = P1 + jQ1) injected by the local power feeder. The four nonlinear equations are

z

n

P 2 = V2

Vj G2j cos θ2j + B2j sin θ2j j=1 n

P 3 = V3

Vj G3j cos θ3j + B3j sin θ3j j=1 n

Vj G2j sinθ2j −B2j cos θ2j

Q2 = V 2 j=1

z

THE POWER FLOW PROBLEM

391

n

Q3 = V 3

Vj G3j sin θ3j −B3j cosθ3j j=1

The above expressions present the basic concepts of bus active and reactive power injections of a power grid. If we know the bus-injected power, then we can solve for the load bus voltages. Voltage calculation is an important step in the design of a power grid network. We should understand that Example 7.2 is not the same as Example 7.1. Example 7.1 is not a practical problem because we cannot expect the local power grid to provide the voltage at the point of interconnection of a microgrid or a feeder. However, in Example 7.2, we know the local power grid bus voltage, and our objective is the design of a feeder to provide the rated voltage to its loads.

7.3

THE POWER FLOW PROBLEM

In the design of a power grid, a fundamental problem is the power flow analysis.1–9 The solution of the power flow ensures that the designed power grid can deliver adequate electric energy to the power grid loads at acceptable voltage and frequency (acceptable voltage is defined as the rated load voltage). For example, for a light bulb rated at 50 W and 120 V, the voltage provided across the load should be 120, with a deviation of no more than 5% under normal operating conditions and 10% under emergency operating conditions. In per unit (p.u.) value, we seek to provide 1 p.u. voltage to the loads within the range of 0.95 and 1.05 p.u. That is, once we have specified the schedule of generation to satisfy the system load demand, the solution for bus voltages must be 1 p.u. ± 5%. The acceptable frequency can be ensured if a balance between the system loads and generation is maintained on a second-by-second basis as controlled by the load frequency control system and by automatic generation control system on 4–10 second control signal to units under its control. The frequency deviation is a fraction of a cycle from the rated frequency—60 Hz in the United States and 50 Hz in the rest of the world. The change in frequency affects the motor loads such as that of the induction motors and pumps. If the frequency drops, the speed of the induction motors will drop and result in excessive heating and failure of the induction motors. For example, consider a power system supported by a few diesel generators. If the operating frequency drops due to heavy load demands on the system, the pumps of a diesel generating station slow down; as a result, the diesel generators are not cooled enough. The overheated diesel engines are then removed from service by the system’s override control system, causing a cascade failure of that power system. Therefore, to maintain stable operation, system bus load voltages are maintained at 1 p.u. with a deviation of no more than 10% in emergency operating conditions.

392

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

The terms power flow studies and load flow studies are used interchangeably to refer to the flow of power from the generating units to the loads. For solving a power flow problem, the question is: “Given the system model and the schedule of generation and loads, find the voltage of the load buses.” 7.4

LOAD FLOW STUDY AS A POWER SYSTEM ENGINEERING TOOL

As we discussed in the preceding section, the power grid must be designed to provide the rated voltage to the grid’s loads. We need to calculate bus voltages from the scheduled transmission system, scheduled generation system, and scheduled bus loads. In power grid system planning, the grid is planned based on projected future loads. The main objective of a power flow study is to determine whether a specific system design with a lower cost can produce bus voltages within an acceptable limit. In general, power grid planning entails many studies addressing (1) power generation planning, (2) transmission system planning, and (3) reactive power supply planning. The objective of planning studies is to ensure that all power grids are operating within their operating limits and bus load voltages are within acceptable limits. In the operation of a power grid, the following questions are addressed. Over the next 24 hours, for all scheduled bus loads, transmission systems, transformers, and generation, will the bus voltages be within the limits of their rated values? If a transformer has oil leaks, can the transformer be taken out of service without affecting load voltages? In the sudden loss of a line, can the power grid load demand be satisfied without any lines being overloaded? Load flow studies in power grid and operational planning, outage control, and power system optimization and stability studies are performed to provide the needed answers. In the following section, we present the power flow problem formulation by first introducing bus types in a power grid. 7.5

BUS TYPES

In a power flow problem, several bus types are defined. The three most important types are a load bus, a generator bus, and a swing bus. Figure 7.3 depicts a load bus.

Power grid

Vk ∠ θk

Pk + jQk

Figure 7.3 A load bus.

BUS TYPES

393

A power system bus has four variables. These variables are (1) the active power at the bus, (2) the reactive power at the bus, (3) the voltage magnitude, and (4) the phase angle. For a load bus, the active and reactive power consumptions are given as a scheduled load for a given time. The time can be specified as the day ahead forecasted peak load. If the system is being planned for 10 years ahead, then the forecasted peak load is used at the bus. Figure 7.4 depicts a generator bus. This type of bus is referred to as a constant P-V bus. A P-V (voltage-controlled) bus models a generator bus. For this bus type, the power injected into the bus by the connected generator is given in addition to the magnitude of bus voltage. The reactive power injected into the network and phase angle must be computed from the solution of the power flow problem. However, the reactive power must be within the limit (minimum and maximum) of what the P-V bus can provide. Figure 7.5 depicts a constant PG–QG bus. This bus type represents a generator with known active and reactive power injection into the power system. However, the magnitude of the generator voltage and the phase angle must be computed from the solution of power flow problem. Figure 7.6 defines a swing bus or slack bus. The swing bus is identical to a P-V bus except the bus voltage is set to 1 p.u. and its phase angle to zero. For a swing bus, the net injected active power and reactive power into the network are not known. The generator connected to the swing bus is called a swing generator or slack generator. The function of a swing bus is to balance power consumption and power loss with net injected generated power. The swing bus can also be considered as an infinite bus, that is, it can theoretically

k VGk ∠ θGk

PGk + jQGk

Power grid

Figure 7.4 A constant voltage-controlled (P-V) bus.

k

PGk + jQGk

Vk ∠ θk

Power grid

Figure 7.5 A constant PG–QG bus.

394

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

k

1∠ 0

Power grid

PGk + jQGk

Figure 7.6 Swing bus or slack bus.

provide an infinite amount of power. Hence, the swing bus is considered an ideal voltage source: it can provide an infinite amount of power, and its voltage remains constant. Note that all the above definitions are identical and are used interchangeably. Let us consider the bus type for microgrids of photovoltaic (PV) or wind generating stations. Because the energy captured from the sun or wind source is free, these types of generating units are operated to produce active power. The PV or wind generating station is operating at unity power factor. Therefore, the PV or wind bus can be modeled as given in Figure 7.7. Figure 7.7 depicts the modeling of a PV or wind generating station connected when a microgrid is connected to the local power grid bus. In this model, for the voltage analysis of the microgrid, the PV or wind bus active power generation is given, and the reactive power generation is assumed to be zero. The bus voltage and phase angle are computed from the solution of the power flow problem subject to a minimum and maximum limitation as specified by the modulation index setting of a PV generating station. Therefore, we can summarize that the PV is generating, k, as specified with active power generation as PGk and the bus voltage as Vmin < Vk < Vmax. At the same time, the reactive power to be provided by a PV or wind generator must be within limits (minimum and maximum) of the generating station. Let us now consider the case when a microgrid is separated from the local power grid. In this case, the local microgrid must control its frequency and bus voltages. When the microgrid of a PV and wind generating system is separated from the local power grid, the PV or wind generating bus can be modeled as given in Figure 7.8.

k

Vk ∠ θk

Power grid PGk

Figure 7.7 A photovoltaic or wind generating station bus model.

BUS TYPES

395

Vmin < Vk < Vmax k and Vk ∠ θk

Power grid PGk + jQGk

Figure 7.8 A photovoltaic or wind generating bus model.

In the above model, the magnitude of bus voltage is specified with a minimum and maximum as defined by the modulation index of the inverter; active power generation and reactive powers are also specified. The phase angle and voltage magnitude are to be computed from the power flow solution. However, a PV generating station without a storage system has very limited control over reactive power. For controlling an inverter power factor, a storage system is essential. An inverter with its supporting storage system that can operate as a steam unit and be able to provide active and reactive power is the subject of ongoing research on modeling and inverter control modeling. In case a wind generating station is connected to the microgrid directly, the reactive power injection control is limited within the acceptable voltage range of a connected wind bus. For an isolated microgrid to operate at a stable frequency and voltage, it must always be able to balance its loads and generation. Because the load variation is continuous and renewable energy sources are intermittent, it is essential that a storage system and a fast-acting generating source such as high-speed microturbines and combined heat and power generating station be part of the generation mix of the microgrid. Students are urged to read Chapter 4 again and study the factors that must be considered for the stable operation of a power grid. In a load flow problem, all buses within the network have a designation. In general, the load buses are modeled as a constant P and Q model where the active power, P, and reactive power, Q, are given and bus voltages are to be calculated. It is assumed that power flowing toward loads is represented as a negative injection into the power system network. The generator buses can be modeled as a constant PG and QG or as P-V bus type. The generators inject algebraically positive active and reactive power into the power system network. For the formulation of the power flow problem, we are interested in injected power into the power system network; the internal impedance of generators is not included in the power system model. However, for short-circuit studies, the internal impedance of the generators is included in the system model. The internal impedance limits the fault current flow from the

396

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

generators. For viable power flow, the balance of the system loads and generation must be maintained at all times. This balance can be expressed as n1

n2

PLk + Plosses

PGk =

(7.10)

k=1

k=1

where PGk is the active power generated by generator k, PLk is the active power consumed by the load k, n1 is the number of the system generators, and n2 is the number of system loads. Similarly, n1

n2

QGk = k=1

QLk + Qlosses

(7.11)

k=1

where QGk is the reactive power generated by generator bus k, QLk is the reactive power consumed by load k, n1 is the number of system generators, and n2 is the number of system loads. Let us consider the system depicted in Figure 7.9.

PV station

Transmission system 1

y′13

DC/AC 2

1–2

y′12

y′24

y′12

1–4

1–3

2–4

y′14

V1

V2 y′14 3–4

4

3 y′34 y′34

y′13

V3 S3

y′24 V4

S4

Figure 7.9 The schematic presentation of a three-bus microgrid system.

397

GENERAL FORMULATION OF THE POWER FLOW PROBLEM

In the system given in Figure 7.9, we will need to balance the three-bus power system loads and generation: PG1 + PG2 = PL4 + PL3 + Plosses PG1 + PG2 −PL4 −PL3 − Plosses = 0

(7.12)

QG1 + QG2 = QL4 + QL3 + Qlosses QG1 + QG2 – QL4 – QL3 – Qlosses = 0

(7.13)

In the above formulation, we assume inductive loads consume reactive power Qind > 0 and capacitive loads supply reactive power Qcap < 0. For ensuring the balance between load and generation, we must calculate the active and reactive power losses. However, to calculate power losses, we need the bus voltages. The bus voltages are the unknown values to be calculated from the power flow formulation. This problem is resolved by defining a bus of the power grid—a swing bus and the generator behind it as a swing generator as defined earlier. The swing bus is an ideal voltage source. As an ideal voltage source, it provides both active and reactive power, while the bus voltage remains constant. Therefore, a swing generator is a source of infinite active and reactive power in a power flow problem formulation. The swing bus voltage is set to 1 p.u. and its phase angle as the reference angle set to zero degrees, Vs = 1∠0. With this assignment, the generator behind the swing bus can provide the required power to loads of the power grid, and its voltage is not subject to fluctuations. Of course, in practice, such a constant voltage source with an infinite power source does not exist. However, if the connected loads are much smaller than the power behind a bus, then it can be approximated as an ideal voltage source. The swing bus allows balancing the system loads plus system losses to the system’s supply generation; thus, the balance of energy in the network is ensured.

7.6

GENERAL FORMULATION OF THE POWER FLOW PROBLEM

Let us now formulate the same problem for a network of a power grid. We should keep in mind the following assumptions: a. b. c. d.

The generators are supplying balanced three-phase voltages. The transmission lines are balanced. The loads are assumed to be balanced. The PV or wind generating stations are presented by a PV bus with the bus voltage having a minimum and maximum limit.

Consider the injection model of a power grid given in Figure 7.10.

398

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

Transmission system

I1

2

1

1–2

y′12

y′13

I2

y′24

y′12

1–4

1–3

2–4

y′14

V1

V2 y′14 3–4

4

3 y′34 y′34

y′13 I3

V3

y′24 V4

I4

Figure 7.10 A current injection model for power flow studies.

In Figure 7.10, the current injections at each bus are presented based on the known power injection and the bus voltage that is calculated from the mathematical model of the system. In Figure 7.10, the following definitions are implied: • The bus voltages are actual bus-to-ground voltages per unit. • The bus currents are net injected currents per unit flowing into the transmission system from generators and loads. • All currents are assigned a positive direction to their respective buses. All generators inject positive currents, and all loads inject negative currents. • The i−j is the one-phase primitive impedance, also called the positive sequence impedance between buses i and j. We study sequence impedances in the next chapter. However, the positive sequence impedance is the same balanced line impedance that we have been using in this book. • The yij is the half of the shunt admittance between buses i and j.

z

Representing the series primitive impedance at the lines by their corresponding primitive admittance form where y12 = 1

z1 – 2; y13 = 1 z1 – 3; y14 = 1 z1 – 4; y24 = 1 z2 – 4; y34 = 1 z3 – 4

(7.14)

GENERAL FORMULATION OF THE POWER FLOW PROBLEM

399

where • y1 = y12 + y13 + y14 is the total shunt admittance connected to bus 1 • y2 = y12 + y24 is the total shunt admittance connected to bus 2 • y3 = y13 + y34 is the total shunt admittance connected to bus 3 • y4 = y14 + y24 + y34 is the total shunt admittance connected to bus 4 The power system in Figure 7.10 can be redrawn as Figure 7.11. Assuming the ground bus as the reference bus, Kirchhoff’s current law for each bus (node) gives I1 = V1 y1 + V1 −V2 y12 + V1 − V3 y13 + V1 −V4 y14 I2 = V2 y2 + V2 −V1 y12 + V2 − V4 y24

(7.15)

I3 = V3 y3 + V3 −V1 y13 + V3 − V4 y34 I4 = V4 y4 + V4 −V1 y14 + V4 − V2 y24 + V4 −V3 y34

I2

Transmission system

I1 1

y12

2

y2

y1

y13

y24 y14

V1

3 V3

y3 I3

y34

V2

4 y4

V4 I4

Figure 7.11 The current injection model by using the admittance representation.

400

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

The above equations can be written as I1 = V1 y1 + y12 + y13 + y14 + V2 − y12 + V3 − y13 + V4 − y14 I2 = V1 − y12 + V2 y2 + y12 + y24 + V3 0 + V4 − y24 I3 = V1 − y13 + V2 0 + V3 y3 + y13 + y34 + V4 − y34

(7.16)

I4 = V1 − y14 + V2 − y24 + V3 − y34 + V4 y4 + y14 + y24 + y34 In matrix form I1 I2 I3 I4

=

Y11 Y12 Y13 Y14

V1

Y21 Y22 Y23 Y24

V2

Y31 Y32 Y33 Y34

V3

Y41 Y42 Y43 Y44

V4

(7.17)

where Y11 = y1 + y12 + y13 + y14 ; Y12 = − y12 ; Y13 = − y13 ; Y14 = −y14 Y21 = Y12 ; Y22 = y2 + y12 + y24 ; Y23 = 0; Y24 = − y24 Y31 = Y13 ; Y32 = Y23 ; Y33 = y3 + y13 + y34 ; Y34 = − y34 Y41 = Y14 ; Y42 = Y24 ; Y43 = Y34 ; Y44 = y4 + y14 + y24 + y34

7.7 7.7.1

ALGORITHM FOR CALCULATION OF BUS ADMITTANCE MODEL The History of Algebra, Algorithm, and Number Systems

Khwarizmi is the father of algebra and algorithm. He was born to a Persian Zoroastrian family who converted to Islam. At this time, the House of Wisdom was the first university established in Baghdad in the eighth century, after the destruction of Plato’s Academia. Khwarizmi was the director of the House of Wisdom when he undertook his famous work as the solution of linear and quadratic equations, The Compendious Book on Calculation by Completion and Balancing (https://www.wdl.org/en/item/7462). He presented the first systematic solution of linear and quadratic equations in Arabic. The word “algorithm” is the Latinization of the name describing his method of solving the second-order equation depending on the steps that must be taken to arrive at the correct solution. The word “algebra” is derived from the title of his book “al-Jaber.” The word al-Jaber in Arabic means forcing two sides of equation to become equal. His work was translated into Latin as Algoritmi de Numero Indorum in the twelfth century. The words algebra and algorithm are derived from Al-Khwārizmī’s treatise.10 He wrote On the Calculation with Hindu

ALGORITHM FOR CALCULATION OF BUS ADMITTANCE MODEL

401

Numerals (https://www.storyofmathematics.com/islamic_alkhwarizmi.htm). The development of the number system is based on the work of Brahmagupta (598–665 CE), an Indian mathematician who invented the Indian numeral systems. Brahmagupta introduced the concept of zero as having no debt, addition as gain (+) and subtraction (−) as debt, and gave elegant description such as 1 + 0 = 1, 1 − 0 = 1, 1 × 0 = 0. Khwarizmi’s work is distinct not only from the Babylonian tablets and Brahmagupta’s work on mathematics but also from the Diophantus’ Arithmetica. The genius of the decimal system is the use of zero as the description of “nothing.” The whole arithmetic is developed from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We all understand the concept of having one item and so on till we have nine items. When we want to express 10 items, the symbol zero invented by Indian mathematician Brahmagupta is used to construct “10.” In the Roman numbering system, the symbol “X” is used, and more symbols are introduced as we count. The significance of the development of the decimal number system is probably best described by the French mathematician Pierre-Simon Laplace (1749–1827 CE) who wrote: “It is India that gave us the ingenious method of expressing all numbers utilizing symbols. Each symbol is receiving a value of a position, as well as an absolute value. A profound and important idea, which appears so simple to us now that we ignore its true merit, but its very simplicity. The great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.” In the twelfth century, Latin translations of Khwarizmi work on the Indian numerals are introduced as “the decimal positional number system” to the Western world. J. J. O’Connor and E. F. Robertson wrote in the MacTutor History of Mathematics archive (www-history.mcs.st-and.ac.uk/): “Khwarizmi, work is the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc. to all be treated as ‘algebraic objects.’ A rational number is any number that can be expressed as the quotient or fraction. Irrational numbers, for example, are 2 and π. The decimal expansion of an irrational number continues without end and repeating. The set of rational numbers is countable, and the irrational numbers are uncountable.” Khwarizmi11 opened a new path for the world of mathematics, initiating a new world for mathematics and physics by moving away from the Greek world of geometry. The formulation of the mathematical algebraic system allowed mathematics to be applied to itself. The new formulation of mathematics by itself was a new revolutionary idea. Later, the renowned poet, mathematician, and astronomer Omar Khayyam (1048–1122)12 wrote Treatise on Demonstration of Problems of Algebra (https://www2.stetson.edu/~efriedma/periodictable/html/O.htm), which laid down the principles of algebra. He also developed algorithms for the root

402

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

extraction of arbitrarily high-degree polynomials.10 Since 825 CE, the word algorithm has been used by mathematicians to formulate and solve complex problems.

7.7.2

Bus Admittance Algorithm

We can formalize the formulation of a bus admittance matrix. We formulate an algorithm for the determination of the Ybus matrix and solve power flow problems. The elements of Ybus matrix can be calculated from the following algorithm. Step 1 If i = j, Yii =

y, that is, the

of admittances connected to bus i

Step 2 If i

j and bus i is not connected to bus j, then the element Yij = 0

Step 3 If i

j and bus i is connected to bus j through the admittance yij ,

then the element is Yij = −yij (7.18) The above algorithm can be easily programmed for the solution of a power flow problem encompassing an eastern US power grid. In a more compact form, we can express the bus current injection vector into a power grid as a bus admittance matrix and a bus voltage vector: Ibus = Ybus Vbus where Ibus is bus injected current vector Ybus is bus admittance matrix Vbus is bus voltage profile vector The Ybus matrix model of the power grid is a symmetric, complex, and sparse matrix. The row sum (or column sum) corresponding to each bus is equal to the admittance to the reference bus. If there is no connection to a reference bus, every row sum is zero. For this case, the Ybus matrix is singular, and det [Ybus] = 0, and such a Ybus matrix cannot be inverted. At this time, we should recall that if we formulate the Ybus matrix model for short-circuit studies, we will include the internal impedance of generators and motors. However, for power flow studies, we represent the power grid with injection models. We should also note that in general, a power grid is normally grounded through the capacitance of transmission lines.

THE BUS IMPEDANCE ALGORITHM

7.8

403

THE BUS IMPEDANCE ALGORITHM

From Figure 7.11, the current bus injections are related to bus voltages by the bus admittance matrix as given below: Ibus = Ybus Vbus Vbus = Zbus Ibus Zbus = Ybus

(7.19)

−1

(7.20)

Therefore, the Zbus matrix is the inverse of the Ybus matrix. Now the bus voltage vector is expressed as Zbus, which is the bus impedance matrix, and Ibus is the bus injected current vector. For the system of Figure 7.11, the impedance matrix of Equation (7.20) can be expressed as V1 V2

=

V3 V4

Z11 Z12 Z13 Z14

I1

Z21 Z22 Z23 Z24

I2

Z31 Z32 Z33 Z34

I3

Z41 Z42 Z43 Z44

I4

(7.21)

Example 7.3 For the power grid given in Figure 7.12, compute the bus admittance and bus impedance models. 2

1

3

0.01 Ω

0.02 Ω

G

V2 0.02 Ω

0.01 Ω V1

0.03 Ω

0.01 Ω

V3

4 V4

Figure 7.12 The power grid for Example 7.3.

404

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

Solution Figure 7.12 depicts the power grid for Example 7.3. The admittance matrix is calculated according to Equation (7.16): 1 1 1 1 + + = 300, Y12 = − y12 = − = − 100, 0 01 0 01 0 01 0 01 1 1 = − 100, Y21 = − y21 = − = − 100, Y14 = − y14 = − 0 01 0 01 1 1 1 1 + + = 200, Y23 = − y23 = − = −50, Y22 = y12 + y23 + y24 = 0 01 0 02 0 02 0 02 1 1 = − 50, Y32 = − y23 = − = − 50, Y24 = − y24 = − 0 02 0 02 1 1 1 + = 83 33, Y34 = − y34 = − = − 33 33, Y33 = y32 + y34 = 0 02 0 03 0 03 1 1 = − 100, Y42 = − y24 = − = − 50, Y41 = − y14 = − 0 01 0 02 1 1 1 1 = − 33 33, Y44 = y41 + y42 + y43 = + + = 183 33 Y43 = − y34 = − 0 03 0 01 0 02 0 03 Y11 = y1 + y12 + y14 =

The rest of the elements of the admittance matrix elements are zero if there are no direct connections between the buses: Y11 Y12 Y13 Y14 Ybus =

Y21 Y22 Y23 Y24 Y31 Y32 Y33 Y34 Y41 Y42 Y43 Y44

300 =

− 100

− 100 200 0

− 50

0

−100

−50

−50

83 33

−33 33

− 100 − 50 −33 33 183 33

0 010 0 010 0 010 0 010 −1 = Zbus = Ybus

0 010 0 017 0 015 0 013 0 010 0 015 0 027 0 015 0 010 0 013 0 015 0 017

where Zbus is the bus impedance model for the power grid of Example 7.3.

7.9

FORMULATION OF THE LOAD FLOW PROBLEM

Consider the power grid presented in Figure 7.11. The power flow problem can mathematically be stated as given by a bus admittance matrix: Ibus = Ybus Vbus

(7.22)

FORMULATION OF THE LOAD FLOW PROBLEM

405

The vector of the current injection represents the net injection where the injected current is algebraically a positive injection for power generation and a negative injection for loads. Therefore, if the generation of a bus is larger than the load connected to the bus, then there is a positive net injection into the power grid. Otherwise, it will be negative if there are more loads connected to the bus than the generating power. Therefore, for each bus k, we have Sk = Vk Ik∗

k = 1, 2, …, n

(7.23)

and Ik is the current injection into the power grid at bus k. Therefore, from row k of the Ybus matrix, we have n

Ik =

Ykj Vj

(7.24)

j=1

By substituting Equation (7.24) in Equation (7.23), we have ∗

n

Sk = Vk

k = 1, 2, …,n

Ykj Vk

(7.25)

j=1

For each bus k, we have a complex equation of the form given by Equation (7.25). Therefore, for the n bus system, we have n nonlinear complex equations. where Ykj = Gkj + jBkj, θkj = θk − θj Vj = Vj cos θj + jsin θj Vk = Vk cos θk + jsin θk Ik∗ =

Sk Vk

∗

=

Pk + jQk Vk∗

∗

=

Pk −jQk Vk ∠ −θk

where n is the total number of buses in the power grid network. From the Ybus model, we have the relationship of injected current into the power grid as it relates to the network admittance model, as well as how the power will flow in the transmission system based on the bus voltages. Therefore, for each bus k, based on the bus admittance model, we have the following expressions: P1 − jQ1 = Y11 V1 + Y12 V2 + Y13 V3 + Y14 V4 V1∗

(7.26)

P2 − jQ2 = Y21 V1 + Y22 V2 + Y23 V3 + Y24 V4 V2∗

(7.27)

406

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

P3 − jQ3 = Y31 V1 + Y32 V2 + Y33 V3 + Y34 V4 V3∗

(7.28)

P4 − jQ4 = Y41 V1 + Y42 V2 + Y43 V3 + Y44 V4 V4∗

(7.29)

We can rewrite Equations (7.26)–(7.29) and express them as P1 −jQ1 = Y11 V12 + Y12 V1∗ V2 + Y13 V1∗ V3 + Y14 V1∗ V4 P2 −jQ2 = Y21 V1 V2∗ + Y22 V22 + Y23 V2∗ V3 + Y24 V2∗ V4 P3 −jQ3 = Y31 V1 V3∗ + Y32 V2 V3∗ + Y33 V32 + Y34 V3∗ V4

(7.30)

P4 −jQ4 = Y41 V1 V4∗ + Y42 V2 V4∗ + Y43 V3 V4∗ + Y44 V42 The above systems of equations are complex and nonlinear. As we stated, one bus of the system is selected as a swing bus, and its voltage magnitude is set to 1 p.u.; its phase angle is set to zero as the reference phasor. The swing bus ensures the balance of power between the system loads and the system generations. In a power flow problem, the load bus voltages are the unknown variables, and all the injected powers are known variables. Here, there are three complex nonlinear equations to be solved for bus load voltages. In general, as we discussed before, for each bus k, a complex equation can be written as two equations in real numbers. Using the above expressions in general formulation, we have n

Pk = Vk

Vj Gkj cos θkj + Bkj sinθkj j=1 n

Qk = V k

Vj Gkj sinθkj − Bkj cos θkj j=1

where Ykj = Gkj + jBkj, θkj = θk − θj Vk = Vk cos θk + jsin θk Vj = Vj cos θj + jsinθj Ik =

Sk Vk

∗

=

Pk − jQk Pk −jQk = ∗ Vk Vk ∠ −θk

THE GAUSS–SEIDEL YBUS ALGORITHM

407

If the system has n bus, the above equations can be expressed as 2n equations: f1 V1 , …, Vn ; θ1 , …, θn = 0 f2 V1 , …, Vn ; θ1 , …, θn = 0 (7.31) fn V1 , …, Vn ; θ1 ,…, θn = 0 f2n V1 ,…, Vn ; θ1 , …,θn = 0 The above 2n equations can be expressed as f1 x f2 x F x =

(7.32) f2n x

where the elements of vector X represent the magnitude of the voltage and phase angle. In Equation (7.32), we have 2n nonlinear equations to be solved, and vector X can be presented as X t = V1 ,…, Vn ; θ1 , …,θn = x1 ,…, xn , xn + 1 , …,x2n

7.10 THE GAUSS–SEIDEL YBUS ALGORITHM The Gauss–Seidel algorithm is an iterative process. In this method, the objective is to satisfy the set of nonlinear equations by repeated approximation. The solution is reached when all nonlinear equations are satisfied at an acceptable accuracy level. In the case of power flow problems, the solution is reached when all bus voltages are converging to around 1 p.u. within 5% of the rated voltage and all nonlinear equations are satisfied at an acceptable tolerance. Let us now restate the main equations of power flow problems:

where k = 1, 2, 3, …, n.

V1 = 1 ∠ 0

(7.33)

Ibus = Ybus Vbus

(7.34)

Sk = Vk Ik∗

(7.35)

408

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

Equation (7.33) is the model for the swing bus (also called a slack bus) that creates a balance between the system loads and system generation by providing the system transmission losses. When a microgrid is connected to a local power grid, the power grid bus is selected as the swing bus; this ensures that the balance between loads and local generation and losses are maintained. If the microgrid has deficiencies in a power generation, the balance is maintained by the power grid bus that acts as an ideal bus or swing bus. If the microgrid has an excess generation, the balance is injected into the local power grid. Equation (7.34) describes the current flow or power flow through transmission lines for a set of bus voltages. Equation (7.35) presents the net injected power on each bus of the system. The Gauss–Seidel Ybus algorithm can be stated as depicted in Figure 7.13. In the Gauss–Seidel method, we repeatedly solve the fundamental load flow equation expressed by Equation (7.31): Pk −jQk − Vk∗ Vk =

n

Ykj Vj j=1 j k

k = 2, …, n

Ykk

(7.36)

Start ↓ V1 = 1 ∠ 0 n

Calculate

Ykj Vj, k = 1,....., n j=1 j≠k

↓ Pk − jQk Calculate update Vk

Vk*

n

YkjVj,

− j=1 j≠k

Vk =

k = 2,....., n

Ykk ↓

Next

bus

until

last

↓ ∆Vk

Tol

∆Pk = Pk (scheduled)−Pk (calculated) ≤ cP

→ Stop

∆Qk = Qk (scheduled)−Qk (calculated) ≤ cQ

Figure 7.13 The Gauss–Seidel algorithm for iterative approximation.

THE GAUSS–SEIDEL YBUS ALGORITHM

409

Equation (7.36) represents the load bus voltages with the power grid depicted by the bus admittance matrix. The diagonal element for a bus for which a voltage approximation is being computed appears in the denominator. Because the diagonal elements of the bus admittance matrix are never zero, an approximation of bus voltages is assured. If the power grid is correctly designed, the convergence can be obtained. In Equation (7.37), we check to see if the original nonlinear load flow equations are satisfied with the last approximated bus voltages: ΔPk = Pk scheduled − Pk calculated ≤ cP ΔQk = Qk scheduled −Qk calculated ≤ cQ

(7.37)

Example 7.4 For the system given in Figure 7.14, use the Gauss–Seidel Ybus method and solve for the bus voltages. For example depicted in Figure 7.14, bus 1 is the swing bus, and its voltage is V1 = 1∠0. The scheduled power at bus 2 is 1.2 p.u. The load at bus 3 is 1.5 p.u. Compute the bus 2 and bus 3 voltages. Solution To solve the above problem, we need to formulate the bus admittance matrix:

Ybus =

14

− 4 − 10

−4

9

−5

−10 − 5 15

2

1 0.25

G

G P2(scheduled) = 1.2

0.1

0.2

3

1.5

Figure 7.14 The one-line diagram of Example 7.4.

410

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

The power flow models are V1 = 1 ∠ 0 Ibus = Vbus Ybus For a DC system, we only have active power flow: Ik∗ = Ik , Vk∗ = Vk Sk = Vk Ik∗ = Vk Ik = Pk , and Qk = 0 Therefore, for bus 2, we have 1 2 = V 2 I2 For bus 3, we have −1 5 = V3 I3 We can start the iterative approximation for i = 0 iterations by assuming bus 2 and bus 3 voltages are equal to 1 p.u.: 0

Vbus =

1 1

For bus 2, we have 3

Y2j Vj = Y21 V1 + Y23 V3 = − 4 1 + −5 1 j=1 j 2

Next, we update V2: P2 −jQ2 − V2 V2 = V2 =

n

Y2j Vj j=1 j 2

Y22 1 12 − Y22 V2

,

i = 1, …, n j

−4 1 0 + − 5 1

i

THE GAUSS–SEIDEL YBUS ALGORITHM

411

The updated bus 2 voltage is given as 1

V2 =

1 12 + 4 + 5 = 1 1333 p u 9 1

We continue the iterative process and update bus 3 voltage: 3

Y3j Vj = Y31 V1 + Y32 V2 j=1 j 3

We update the bus 3 voltage, V3:

V3 =

1 −1 5 − Y31 V1 + Y32 V2 Y33 V3

V3 =

1 −1 5 − 15 1 0

1

V3 =

− 10 1 + −5 1 1333

1 14 1666 −1 5 + 10 + 5 666 = = 0 9444 p u 15 15

We continue the approximation by calculating the mismatch at bus 2 and bus 3. The mismatch at bus 2 is 3

P2 calculated =

V2 I2j j=0 j 2

P2 calculated = V2 I20 + V2 I21 + V2 I23 P2 calculated = 0 + 1 1333

V2 − V1 V2 − V3 + 1 333 0 25 02

P2 calculated = 0 + 1 1333

1 1333 −1 0 1 1333 −0 944 + 1 1333 = 1 6769 0 25 02

ΔP2 = P2 scheduled −P2 calculated = 1 2 − 1 6769 ΔP2 = −0 4769 p u

412

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

TABLE 7.1 Example 7.4 Results Bus

Voltage (p.u.)

Power Mismatch (p.u.)

1.078 0.917

0.63 × 10−4 0.28 × 10−4

2 3

The mismatch at bus 3 is 3

P3 calculated =

V3 I3j j=0 j 3

P3 calculated = V3 I30 + V3 I31 + V3 I32 P3 calculated = 0 + 0 944 P3 calculated = 0 944

V3 − V1 V3 − V2 + 0 944 01 02

0 944 −1 0 0 944− 1 1333 + 0 944 = −1 4221 01 02

ΔP3 = P3 scheduled − P3 calculated = − 1 5 − −1 4221 ΔP3 = −0 0779 p u The process is continued until error reduces to a satisfactory value. The result is obtained after seven iterations with cp of 1 × 10−4. The results are given in Table 7.1. The power supplied by bus 1, the swing bus, is equal to the total load minus total generation by all other buses plus the losses.: 3

P 1 = V 1 I1 = V 1

Y1j Vj = V1 Y11 V1 + Y12 V2 + Y13 V3 j=1

The bus voltage of swing bus is 1 ∠ 0 and the p.u. power injected by bus is P1 = 0.522 p.u. The total power loss of the transmission lines is 0.223 p.u.

7.11 THE GAUSS–SEIDEL ZBUS ALGORITHM In the Gauss–Seidel Zbus algorithm method,6 the power problem can be expressed as V1 = 1∠0, which defines the swing bus voltage: Vbus = Zbus Ibus

THE GAUSS–SEIDEL ZBUS ALGORITHM

413

Gauss ↓ V1 = 1∠0 n

V1−∑ Z1j Ij

Calculate

Update I1

j=2

I1 =

Z11

↓

Calculate

n

Updates of Vj and Ij interleaved

V1 = ∑ Z2j Ij j=1

`` ``

↓ S2 * I2 = V2 ↓ n

`` . . . .

V3 = ∑ Z3 j Ij j=1

↓ S3 * I3 = V3 ↓

`` ``

Next

bus

until

last

↓ k = 1,2,.....,n Stopping ∆Vk Tol ∆Pk = Pk (scheduled) − Pk (calculated) ≤ cP

criteria

→ Stop

∆Qk = Qk (scheduled) − Qk (calculated) ≤ cQ

Figure 7.15 The Gauss–Seidel Zbus algorithm.

The Zbus defines the power flow through the transmission system: Sk = Vk Ik∗ The Zbus defines the injection model. The Gauss–Seidel Zbus algorithm can be depicted in Figure 7.15 as an iterative approximation as summarized below. Example 7.5 Consider the system shown in Figure 7.16. Find the bus voltages using the Gauss–Seidel Zbus algorithm. Bus 1 is the swing bus, and its voltage is set to V1 = 1∠0. Without the fictitious line to ground, the Zbus is not defined. The swing bus is grounded, and power drawn by this bus will be accounted for when the problem is solved. The tie to the ground is selected in the same order as the line impedances. The injected power is S2 = −½, S3 = −1, and S4 = −½. All data are expressed in per unit.

414

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

3

2

1 0.01

0.02 1

G

1/2

Fictitious tie

0.01 0.03

0.01 V2 V1

V3

4 1/2

V4

Figure 7.16

The one-line diagram of Example 7.5.

Solution The Zbus with respect to the ground bus is 0 01 Zbus =

0 01

0 01

0 01

0 01 0 0186 0 0157 0 0114 0 01 0 0157 0 0271 0 0143 0 01 0 0114 0 0143 0 0186

The power flow problem can be expressed as V1 = 1 ∠ 0 Vbus = Zbus Ibus For a DC system, we only have active power flow: Sk = Vk Ik∗ = Vk Ik = Pk , Qk = 0 The bus load and generation can be expressed as −0 5 = V2 I2 −1 0 = V3 I3 −0 5 = V4 I4

THE GAUSS–SEIDEL ZBUS ALGORITHM

415

The first step is to compute the Gauss zero iteration (0) values for bus voltages and current as given below: 1 0

Vbus =

0 0

0 0

, Ibus =

0

0 0 0

For the first row of Vbus = ZbusIbus, we have the following: By updating I1, n

Z1j Ij

V1 − j=2

I1 =

Z11

=

1 −0 = 100 0 01

n=4

By updating V2, n

V2 =

Z2j Ij = 0 01 100 + 0 0186 0 + 0 0157 0 + 0 0114 0 = 1 j=1

By updating I2, I2 =

S2 = V2

−0 5 = −0 5 1

By updating V3, n

V3 =

Z3j Ij = 0 01 100 + 0 0157 − 0 5 + 0 0271 0 + 0 0143 0 j=1

= 1 − 0 0078 = 0 9922 By updating I3, I3 =

S3 −1 = − 1 0079 = 0 9922 V3

By updating V4, n

V4 =

Z4j Ij = 0 01 100 + 0 0114 −0 5 + 0 0143 − 1 0079 + 0 0186 0 j=1

= 1 − 0 0057 − 0 0144 = 0 9799

416

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

By updating I4, I4 =

S4 −0 5 = = −0 5102 0 9799 V4

From the above calculation, we have 1 1

Vbus =

1 0 9922

100 −0 5

1

, Ibus =

0 9799

−1 0079 −0 5102

By updating I1, I1 =

1 − 0 01 − 0 5 − 0 01 −1 0079 − 0 01 −0 5102 1 02019 = = 102 019 0 01 0 01

By updating V2, V2 = 0 01 102 019 + 0 0186 − 0 5 + 0 0157 − 1 0079 + 0 0114 − 0 5102 = 0 9892

By updating I2, I2 =

S2 −0 5 = = −0 5055 0 9892 V2

By updating V3, V3 = 0 01 102 019 + 0 0157 − 0 5055 + 0 0271 − 1 0079 + 0 0143 −0 5102 = 0 9776 By updating I3, I3 =

S3 −1 = = −1 023 0 9776 V3

By updating V4, n

V4 =

Z4j Ij = 0 01 102 019 + 0 0114 − 0 5055 + 0 0143 −1 023 j=1

+ 0 0186 − 0 5102 = 0 9903

THE GAUSS–SEIDEL ZBUS ALGORITHM

417

By updating I4, I4 =

S4 −0 5 = = − 0 5049 0 9903 V4

The mismatch at each bus k is given as P calculated k = Vk Ik ΔPk = P scheduled k −P calculated k For bus 2, P2 calculated = V2 I2 ΔP2 = P2 scheduled − P2 calculated For bus 3, P3 calculated = V3 I3 ΔP3 = P3 scheduled − P3 calculated The process is continued until error reduces to a satisfactory value. The results are obtained after four iterations with cp of 1 × 10−4. The results are provided in Table 7.2. If the error is more than that for satisfactory performance, the next iteration is followed.

The MATLAB simulation testbed for the above problem is given below. %Power Flow: Gauss-Seidel method clc; clear all; tolerance= 1e-4; N=4; % no. of buses

TABLE 7.2 Example 7.5 Results Bus 2 3 4

Voltage (p.u.)

Power Mismatch (p.u.)

0.99 0.98 0.99

0.1306 × 10−5 0.2465 × 10−5 0.6635 × 10−5

418

LOAD FLOW ANALYSIS OF POWER GRIDS AND MICROGRIDS

Y=[1/0.01+1/0.01+1/0.01 -1/0.01 0 -1/0.01; -1/0.01 1/0.01+1/0.02 -1/0.02 0; 0 -1/0.02 1/0.02+1/0.03 -1/0.03; -1/0.01 0 -1/0.03 1/0.01+1/0.03]; Z=inv(Y) P_sch=[1 - 0.5 - 1 - 0.5]' I=[0 0 0 0]'; V=[1 0 0 0]'; iteration=0; while (iteration